#7595755

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(17,18,22,27)(19,26,30,24)(20,32,36,48)(21,38,31,41)(23,39,34,28)(25,42,44,37)(29,43,46,47)(33,40,35,45);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,26)(20,33,36,35)(21,34,31,23)(25,43,44,47)(28,38,39,41)(29,37,46,42)(32,45,48,40);(10,14)(11,16)(17,20,34,42,22,36,23,37)(18,25,39,48,27,44,28,32)(19,29,21,33,30,46,31,35)(24,40,38,43,26,45,41,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,19,22,31,34,30)(18,26,28,38,27,24,39,41)(20,35,37,46,36,33,42,29)(25,40,32,47,44,45,48,43);(2,5)(6,8)(10,14)(13,15)(18,27)(20,36)(24,26)(28,39)(29,46)(33,35)(37,42)(38,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=fgi,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^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, 7502951);