#7410110

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1262 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)(4,7)(5,8)(9,10,13,12)(11,14,15,16)(17,18)(19,29)(20,26)(21,28)(22,24)(23,27)(25,31)(30,43)(32,42)(33,38)(34,41)(35,40)(36,37)(39,46)(44,48)(45,47);(1,3,6,2)(4,7,5,8)(9,11)(10,14)(12,16)(13,15)(18,24)(20,33)(25,37)(27,28)(29,42)(30,44)(34,35)(45,46);(10,12)(11,15)(17,19,22,32)(18,25,24,37)(20,30,33,44)(21,31,23,36)(26,39,38,47)(27,29,28,42)(34,45,35,46)(40,48,41,43);(1,4,6,5)(2,7,3,8)(9,12,13,10)(11,14,15,16)(17,20)(18,26)(19,30)(21,35)(22,33)(23,34)(24,38)(25,39)(27,41)(28,40)(29,43)(31,46)(32,44)(36,45)(37,47)(42,48);(1,5,6,4)(2,7,3,8)(11,15)(14,16)(17,21,22,23)(18,27,24,28)(19,31,32,36)(20,34,33,35)(25,29,37,42)(26,40,38,41)(30,45,44,46)(39,48,47,43)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,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, 7317306);