#7512007

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1536 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419203);