#7043893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6951089);