#94003

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:

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

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=efhi,c^b=ch,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, 1199);