#7595112

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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18,24,30)(19,25,33,26)(20,34,32,42)(21,36,40,45)(22,28,23,29)(27,39,46,38)(31,47,48,44)(35,41,37,43);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,33)(18,25,30,26)(20,22,32,23)(21,37,40,35)(27,44,46,47)(28,42,29,34)(31,38,48,39)(36,43,45,41);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,23,19,24,32,22,33)(18,26,28,42,30,25,29,34)(21,35,39,31,40,37,38,48)(27,43,45,47,46,41,36,44);(9,11,12,16)(10,14,13,15)(17,21,23,39,24,40,22,38)(18,27,28,45,30,46,29,36)(19,31,32,37,33,48,20,35)(25,41,34,44,26,43,42,47);(2,5)(6,8)(17,22,24,23)(18,28,30,29)(19,32,33,20)(21,38,40,39)(25,42,26,34)(27,45,46,36)(31,37,48,35)(41,47,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=gi,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=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, 7502308);