#7487812

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 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)(19,23)(20,25)(21,24)(22,27)(26,33)(28,36)(29,35)(30,37)(31,40)(32,38)(34,42)(39,44)(41,47)(43,46)(45,48);(2,5)(3,7)(10,14)(11,12)(17,19)(18,23)(20,29)(21,30)(22,28)(24,37)(25,35)(26,34)(27,36)(31,41)(32,43)(33,42)(38,46)(39,45)(40,47)(44,48);(9,10)(11,16)(12,15)(13,14)(18,24)(19,28)(20,31)(21,33)(23,34)(25,39)(26,27)(35,46)(36,37)(38,40)(43,48)(45,47);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,31)(18,25,27,40)(19,29,28,41)(21,32,33,44)(23,35,36,47)(24,38,26,39)(30,43,42,48)(34,45,37,46);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(19,28)(20,31)(23,36)(25,40)(30,42)(32,44)(34,37)(38,39)

Polycyclic presentation:

2^9|d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7395008);