#7595362

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,29,20,24)(21,35,38,48)(22,39,33,27)(25,32,41,31)(26,43,45,37)(30,42,34,44)(36,46,47,40);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,23)(21,36)(22,31)(25,27)(26,44)(28,29)(30,37)(32,33)(34,43)(35,40)(38,47)(39,41)(42,45)(46,48);(10,14)(11,16)(17,20,33,32,23,19,22,31)(18,25,39,29,28,41,27,24)(21,34,43,36,38,30,37,47)(26,42,48,46,45,44,35,40);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,33,43,23,38,22,37)(18,26,39,48,28,45,27,35)(19,30,31,47,20,34,32,36)(24,40,25,42,29,46,41,44);(2,5)(6,8)(10,14)(13,15)(17,22,23,33)(18,27,28,39)(19,31,20,32)(21,37,38,43)(24,25,29,41)(26,35,45,48)(30,47,34,36)(40,42,46,44)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=fg,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,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, 7502558);