#7614453

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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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)(15,23)(16,27)(17,30)(19,33)(20,28)(21,32)(24,35)(25,29)(26,40)(31,36)(34,37)(38,43);(2,4)(6,10)(11,12)(13,14,18,22)(15,24,26,33)(16,28,31,43)(17,32,30,21)(19,23,35,40)(20,36,38,27)(25,34,29,37)(39,41,42,44);(5,7,9,8)(6,11,10,12)(13,15,17,23,18,26,30,40)(14,19,21,33,22,35,32,24)(16,29,27,42,31,25,36,39)(20,37,28,41,38,34,43,44);(1,2,3,4)(13,16,30,36,18,31,17,27)(14,20,32,43,22,38,21,28)(15,25,40,42,26,29,23,39)(19,34,24,41,35,37,33,44);(5,8,9,7)(6,11,10,12)(16,31)(20,38)(25,29)(27,36)(28,43)(34,37)(39,42)(41,44)

Polycyclic presentation:

2^9|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, 7521649);