#7230526

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,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, 7137722);