#140559

order	512
nilpotency class	4
isoclinism	isoclinic class of #13215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1961
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9642
prime radical	#3147	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,12,8,14,31,43,9,15,30,44,6,13,5,29)(3,16,20,35,19,46,53,37,21,48,55,36,32,47,18,10)(4,22,26,58,27,11,24,40,28,41,33,57,25,42,61,39)(17,49,51,34,23,59,50,63,52,64,54,56,60,45,62,38);(1,3,8,19,9,21,6,32)(2,10)(4,23,25,17,28,60,27,52)(5,18,7,20,31,53,30,55)(11,38,42,63)(12,16)(13,36)(14,35)(15,37)(22,56,41,34)(24,54,26,50,61,51,33,62)(29,47)(39,64,40,49)(43,46)(44,48)(45,57,59,58);(1,4)(2,11,15,42)(3,17,21,52)(5,24)(6,25)(7,33)(8,27)(9,28)(10,34)(12,39,44,40)(13,22,14,41)(16,45)(18,50,53,62)(19,23,32,60)(20,54,55,51)(26,30)(29,57,43,58)(31,61)(35,63)(36,38)(37,56)(46,49)(47,64)(48,59)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47755);