#7610568

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

1448 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)(19,24)(20,32)(21,35)(22,27)(23,28)(25,31)(26,38)(29,47)(30,42)(33,40)(34,43)(36,44)(37,46)(39,45)(41,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,30)(18,24,28,42)(20,33,31,22)(21,36,38,34)(25,27,32,40)(26,43,35,44)(29,37,48,39)(41,45,47,46);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,25)(19,22)(21,36)(23,31)(24,40)(26,44)(27,42)(28,32)(29,37)(30,33)(34,38)(35,43)(39,48)(41,46)(45,47);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,33,37,23,38,22,39)(18,26,40,45,28,35,27,46)(19,29,20,34,30,48,31,36)(24,41,32,44,42,47,25,43);(2,8)(6,7)(17,22,23,33)(18,27,28,40)(19,20,30,31)(21,37,38,39)(24,32,42,25)(26,45,35,46)(29,36,48,34)(41,43,47,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fhi,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,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, 7517764);