#7595593

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

1534 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)(19,24,30,25)(20,31,34,40)(21,35,36,48)(22,38,32,27)(26,42,44,37)(29,43,45,46)(33,39,47,41);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,28,25)(20,22,34,32)(21,33,36,47)(26,43,44,46)(27,31,38,40)(29,42,45,37)(35,39,48,41);(10,14)(11,16)(17,20,32,19,23,34,22,30)(18,25,38,40,28,24,27,31)(21,33,42,29,36,47,37,45)(26,41,48,43,44,39,35,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,37,23,36,32,42)(18,26,27,35,28,44,38,48)(19,29,20,33,30,45,34,47)(24,39,40,43,25,41,31,46);(2,5)(6,8)(10,14)(13,15)(21,36)(26,44)(29,45)(33,47)(35,48)(37,42)(39,41)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=fg,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^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7502789);