#7610664

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(21,26)(22,27)(23,28)(25,31)(29,39)(30,40)(33,38)(34,46)(35,41)(36,42)(37,43)(44,47)(45,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,30)(18,24,28,40)(20,33,31,22)(21,35,37,34)(25,27,32,38)(26,41,43,46)(29,44,45,36)(39,47,48,42);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,25)(19,22)(21,34)(23,31)(24,38)(26,41)(27,40)(28,32)(29,36)(30,33)(35,37)(39,47)(42,48)(43,46)(44,45);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,33,44,23,37,22,36)(18,26,38,47,28,43,27,42)(19,29,20,34,30,45,31,35)(24,39,32,46,40,48,25,41);(2,8)(6,7)(17,22,23,33)(18,27,28,38)(19,20,30,31)(21,36,37,44)(24,32,40,25)(26,42,43,47)(29,34,45,35)(39,46,48,41)

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=dgi,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, 7517860);