#7361732

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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)(17,18,21,26)(19,28,29,23)(20,32,33,25)(22,24,35,27)(30,37,42,41)(31,36,43,38)(34,40,45,39)(44,47,48,46);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,30)(21,29)(22,31)(24,38)(25,37)(26,28)(27,36)(32,41)(33,42)(34,44)(35,43)(39,47)(40,46)(45,48);(9,11,12,16)(10,13,14,15)(18,24)(19,29)(20,33)(22,35)(23,36)(25,40)(26,27)(28,38)(32,39)(37,47)(41,46)(44,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,21,33)(18,25,26,32)(19,30,29,42)(22,34,35,45)(23,37,28,41)(24,39,27,40)(31,44,43,48)(36,46,38,47);(2,5)(6,8)(10,14)(13,15)(18,26)(19,29)(20,33)(24,27)(31,43)(34,45)(37,41)(46,47)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7268928);