#7064213

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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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)(13,14,17,23,18,22,29,20)(15,19,26,30,24,32,38,33)(16,21,28,35,27,36,40,34)(25,31,39,41,37,43,44,42);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,33)(21,31)(22,32)(23,30)(27,37)(28,39)(29,38)(34,42)(35,41)(36,43)(40,44);(5,7,8,12)(6,9,10,11)(14,20)(15,24)(16,27)(17,29)(19,30)(21,35)(22,23)(31,42)(32,33)(34,36)(39,44)(41,43);(1,3,4,2)(7,12)(9,11)(13,16,18,27)(14,21,22,36)(15,25,24,37)(17,28,29,40)(19,31,32,43)(20,34,23,35)(26,39,38,44)(30,41,33,42);(6,10)(9,11)(14,22)(15,24)(16,27)(20,23)(26,38)(28,40)(31,43)(41,42)

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=fi,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, 6971409);