#7610830

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,25,32,41)(20,26,34,43)(21,35,38,27)(22,28,23,29)(31,47,48,40)(33,44,36,42)(37,46,39,45);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,22)(21,33)(23,34)(24,32)(26,28)(27,42)(29,43)(30,41)(31,37)(35,44)(36,38)(39,48)(40,45)(46,47);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,26)(19,22)(21,36)(23,32)(24,34)(25,28)(27,44)(29,41)(30,43)(31,37)(33,38)(35,42)(39,48)(40,45)(46,47);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,23,37,24,38,22,39)(18,27,29,45,30,35,28,46)(19,31,20,33,32,48,34,36)(25,40,26,42,41,47,43,44);(2,8)(6,7)(17,22,24,23)(18,28,30,29)(19,20,32,34)(21,37,38,39)(25,26,41,43)(27,45,35,46)(31,36,48,33)(40,44,47,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7518026);