#7063085

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	#25131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1853 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)(13,14,19,24)(15,25,16,21)(17,22,32,41)(18,33,35,23)(20,29,37,27)(26,39,28,38)(30,36,43,44)(31,42,34,40);(5,6)(7,11)(8,10)(9,12)(13,15,19,16)(14,20,24,37)(17,30,32,43)(18,29,35,27)(21,33,25,23)(22,39,41,38)(26,31,28,34)(36,40,44,42);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,15)(14,21,24,25)(17,26,32,28)(18,29,35,27)(20,33,37,23)(22,39,41,38)(30,31,43,34)(36,42,44,40);(7,12)(9,11)(13,17,19,32)(14,22,24,41)(15,26,16,28)(18,34,35,31)(20,36,37,44)(21,38,25,39)(23,42,33,40)(27,30,29,43);(6,10)(9,11)(15,16)(20,37)(21,25)(26,28)(27,29)(30,43)(36,44)(38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,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, 6970281);