#7595589

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

1532 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)(17,18,24,30)(19,25,33,26)(20,34,32,42)(21,36,38,46)(22,29,23,28)(27,40,45,39)(31,47,48,44)(35,41,37,43);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,30,26)(20,22,32,23)(21,37,38,35)(27,44,45,47)(28,34,29,42)(31,39,48,40)(36,43,46,41);(10,14)(11,16)(17,20,23,19,24,32,22,33)(18,26,29,42,30,25,28,34)(21,35,40,31,38,37,39,48)(27,43,46,47,45,41,36,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,39,24,38,23,40)(18,27,28,36,30,45,29,46)(19,31,20,35,33,48,32,37)(25,41,42,47,26,43,34,44);(2,5)(6,8)(10,14)(13,15)(21,38)(27,45)(31,48)(35,37)(36,46)(39,40)(41,43)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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, 7502785);