#7561181

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,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^d=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, 7468377);