#7614220

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=hi,e^2=fi,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=cg,d^b=dh,g^b=gi,h^b=hi,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, 7521416);