#7048362

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1306 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fgi,d^2=i,e^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=eh,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, 6955558);