#7595115

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(12,13)(14,16)(17,18,24,29)(19,25,32,41)(20,33,35,44)(21,38,31,40)(22,28,23,27)(26,37,43,36)(30,45,46,42)(34,48,47,39);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,32)(18,25,29,41)(20,34,35,47)(21,23,31,22)(26,42,43,45)(27,40,28,38)(30,36,46,37)(33,48,44,39);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,36,24,35,23,37)(18,26,27,33,29,43,28,44)(19,30,31,47,32,46,21,34)(25,39,38,45,41,48,40,42);(9,11,12,16)(10,14,13,15)(17,21,23,32,24,31,22,19)(18,25,28,40,29,41,27,38)(20,34,37,46,35,47,36,30)(26,39,44,42,43,48,33,45);(2,5)(6,8)(20,35)(26,43)(30,46)(33,44)(34,47)(36,37)(39,48)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=gi,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,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, 7502311);