#7064471

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,23,30,20)(15,19,25,33,26,34,41,31)(16,27,29,35,28,21,36,37)(24,38,40,42,39,32,43,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,23,34)(16,24,28,39)(17,25,30,41)(20,31,22,33)(21,32,27,38)(29,40,36,43)(35,42,37,44);(5,7,8,12)(6,9,10,11)(14,20)(17,30)(19,31)(21,35)(22,23)(25,41)(27,37)(29,36)(32,42)(33,34)(38,44)(40,43);(1,3,4,2)(7,12)(9,11)(13,16)(14,21)(15,24)(17,29)(18,28)(19,32)(20,35)(22,37)(23,27)(25,40)(26,39)(30,36)(31,42)(33,44)(34,38)(41,43);(6,10)(9,11)(16,28)(21,27)(24,39)(29,36)(32,38)(35,37)(40,43)(42,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971667);