#7205693

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18)(19,28)(20,31)(21,26)(22,27)(23,24)(25,29)(30,40)(32,44)(33,42)(34,37)(35,41)(36,39)(38,43)(45,48)(46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,24,27,28)(20,29,33,36)(21,34,35,30)(25,31,39,42)(26,40,41,37)(32,45,46,43)(38,47,48,44);(3,7)(5,8)(10,14)(15,16)(17,20,22,33)(18,25,27,39)(19,29,23,36)(21,32,35,46)(24,31,28,42)(26,38,41,48)(30,43,34,45)(37,44,40,47);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,30)(20,32)(22,35)(23,34)(24,37)(25,38)(27,41)(28,40)(29,43)(31,44)(33,46)(36,45)(39,48)(42,47);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(18,27)(20,33)(21,35)(24,28)(29,36)(30,34)(38,48)(44,47)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^c=ei,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, 7112889);