#7609514

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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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)(11,13)(12,14)(15,16)(17,18,24,30)(19,25,33,41)(20,26,32,40)(21,27,38,46)(22,28,23,29)(31,39,47,48)(34,42,35,43)(36,44,37,45);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,33)(18,25,30,41)(20,22,32,23)(21,35,38,34)(26,28,40,29)(27,43,46,42)(31,36,47,37)(39,44,48,45);(10,14)(11,15)(17,20,24,32)(18,26,30,40)(19,22,33,23)(21,34,38,35)(25,28,41,29)(27,42,46,43)(31,36,47,37)(39,44,48,45);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,37,24,38,22,36)(18,27,29,45,30,46,28,44)(19,31,32,35,33,47,20,34)(25,39,40,43,41,48,26,42);(2,5)(6,8)(17,22,24,23)(18,28,30,29)(19,32,33,20)(21,36,38,37)(25,40,41,26)(27,44,46,45)(31,35,47,34)(39,43,48,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,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, 7516710);