#7064534

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 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)(13,14,19,25,20,26,18,24)(15,21,29,37,30,38,34,40)(16,23,31,39,17,32,33,22)(27,36,42,44,28,43,41,35);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,27)(17,28)(18,34)(19,29)(20,30)(22,35)(23,36)(24,40)(25,37)(26,38)(31,42)(32,43)(33,41)(39,44);(5,7,8,12)(6,9,10,11)(13,16,20,17)(14,22,26,39)(15,27,30,28)(18,31,19,33)(21,35,38,44)(23,25,32,24)(29,41,34,42)(36,37,43,40);(1,3,4,2)(7,12)(9,11)(13,17,20,16)(14,23,26,32)(15,28,30,27)(18,31,19,33)(21,36,38,43)(22,25,39,24)(29,41,34,42)(35,37,44,40);(6,10)(9,11)(13,18,20,19)(14,24,26,25)(15,29,30,34)(16,31,17,33)(21,37,38,40)(22,23,39,32)(27,41,28,42)(35,43,44,36)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971730);