#7614454

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1807 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)(7,11)(8,12)(9,10)(13,14,19,22)(15,20,25,37)(16,18,28,31)(17,27,32,30)(21,38,40,33)(23,39,42,43)(24,26,34,36)(29,35,44,41);(2,4)(6,10)(11,12)(13,15,19,25)(14,20,22,37)(16,26,28,36)(17,29,32,44)(18,34,31,24)(21,39,40,43)(23,38,42,33)(27,35,30,41);(5,7,9,8)(6,11,10,12)(13,16,18,22,19,28,31,14)(15,20,24,36,25,37,34,26)(17,30,33,21,32,27,38,40)(23,41,29,43,42,35,44,39);(1,2,3,4)(13,17,31,38,19,32,18,33)(14,21,28,30,22,40,16,27)(15,23,34,44,25,42,24,29)(20,35,26,43,37,41,36,39);(5,8,9,7)(6,11,10,12)(17,32)(21,40)(23,42)(27,30)(29,44)(33,38)(35,41)(39,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=hi,e^2=f,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, 7521650);