#7063077

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	#23975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1860 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

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, 6970273);