#7516769

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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)(14,20)(15,25)(16,30)(17,19)(21,36)(22,38)(26,34)(28,29)(31,35)(32,33)(37,43)(39,41);(1,2)(3,4)(6,9)(10,12)(13,14,18,20)(15,26,29,36)(16,31,32,38)(17,24,19,23)(21,28,34,25)(22,33,35,30)(27,37,42,43)(39,44,41,40);(13,15,18,29)(14,21,20,34)(16,27,32,42)(17,25,19,28)(22,37,35,43)(23,26,24,36)(30,39,33,41)(31,40,38,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,32)(14,22,20,35)(15,27,29,42)(17,30,19,33)(21,37,34,43)(23,31,24,38)(25,39,28,41)(26,40,36,44);(6,9)(10,12)(13,17,18,19)(14,23,20,24)(15,28,29,25)(16,30,32,33)(21,36,34,26)(22,31,35,38)(27,41,42,39)(37,44,43,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=i,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,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423965);