#7308832

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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

1441 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=f,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,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, 7216028);