#7308695

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	#2407	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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,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, 7215891);