#7064232

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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1838 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)(13,14,19,31,18,24,17,23)(15,20,32,42,27,36,26,35)(16,22,28,21,30,37,29,38)(25,34,39,33,41,43,40,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,27)(14,20,24,36)(16,25,30,41)(17,26,19,32)(21,33,38,44)(22,34,37,43)(23,35,31,42)(28,39,29,40);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,28)(18,30)(19,29)(20,33)(22,31)(23,37)(24,38)(26,39)(27,41)(32,40)(34,42)(35,43)(36,44);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,25)(17,29)(18,30)(19,28)(20,34)(21,31)(23,38)(24,37)(26,40)(27,41)(32,39)(33,42)(35,44)(36,43);(6,10)(9,11)(13,17,18,19)(14,23,24,31)(15,26,27,32)(16,28,30,29)(20,35,36,42)(21,37,38,22)(25,39,41,40)(33,43,44,34)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971428);