#94004

order	512
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	H(3,ℤ/4)
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	#2108	C₂²
socle	C₂	#7144
prime radical	C₂³	#350
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ G
LECS	G ▷ #2108 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

112 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2,5,12,7,14,31,52,8,15,32,53,37,58,86,111,9,16,33,54,38,59,87,112,6,13,30,51,35,56,84,109)(3,17,19,60,21,63,68,117,20,62,29,81,70,105,82,123,22,61,69,116,36,89,91,125,34,44,83,97,88,98,85,124)(4,11,24,45,27,48,75,101,28,49,76,102,80,106,122,128,25,46,73,99,77,103,119,126,26,47,74,100,78,104,120,127)(10,39,40,79,42,93,94,65,41,92,50,90,96,121,107,23,43,71,95,72,57,114,115,18,55,64,108,67,113,118,110,66);(1,3)(2,10)(4,23,25,66)(5,29)(6,34)(7,36)(8,20)(9,22)(11,44,46,62)(12,50)(13,55)(14,57)(15,41)(16,43)(17,47,61,49)(18,26,65,28)(19,30)(21,38)(24,64,73,92)(27,79,77,72)(31,85)(32,69)(33,83)(35,70)(37,88)(39,74,71,76)(40,51)(42,59)(45,60,99,116)(48,105,103,98)(52,110)(53,95)(54,108)(56,96)(58,113)(63,106,89,104)(67,80,90,78)(68,86)(75,121,119,118)(81,100,97,102)(82,87)(84,91)(93,122,114,120)(94,111)(101,125,126,117)(107,112)(109,115)(123,128,124,127)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,f^2=i,g^2=h,h^2=i,b^a=bc,e^a=efh,c^b=chi,d^b=df,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1200);