#7595626

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

Polycyclic presentation:

2^9|a^2=i,b^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=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, 7502822);