#7207276

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,24,34,43)(20,35,37,26)(21,28,22,27)(25,38,44,36)(30,40,31,45)(32,42,33,41)(39,47,48,46);(1,3,6,2)(4,8,5,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,32)(22,33)(23,34)(25,39)(26,45)(27,41)(28,42)(29,43)(31,37)(35,40)(36,46)(38,47)(44,48);(1,4,6,5)(2,7,3,8)(11,15)(13,16)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,28)(27,35)(29,44)(31,34)(32,46)(33,47)(40,41)(42,45)(43,48);(1,4,6,5)(2,8,3,7)(10,14)(11,15)(17,20)(18,26)(19,31)(21,38)(22,36)(23,37)(24,40)(25,28)(27,44)(29,35)(30,34)(32,46)(33,47)(39,41)(42,48)(43,45);(1,5,6,4)(2,8,3,7)(11,15)(13,16)(17,21,23,22)(18,27,29,28)(19,32,34,33)(20,36,37,38)(24,41,43,42)(25,35,44,26)(30,46,31,47)(39,40,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,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, 7114472);