#7020062

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	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=i,e^2=i,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, 6927258);