#7312298

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#53152	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 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)(17,18,23,35,22,26,21,27)(19,28,34,24,31,41,30,38)(20,25,36,46,32,40,33,39)(29,42,45,37,43,48,44,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,38)(20,29,32,43)(21,34,23,30)(25,37,40,47)(27,41,35,28)(33,45,36,44)(39,48,46,42);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,32)(18,25,26,40)(19,29,31,43)(21,33,23,36)(24,37,38,47)(27,39,35,46)(28,42,41,48)(30,44,34,45);(2,5)(6,8)(10,14)(13,15)(18,26)(20,32)(24,38)(27,35)(28,41)(29,43)(33,36)(44,45)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7219494);