#7561217

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1827 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7468413);