#6850108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,31)(21,35)(22,28)(23,24)(25,32)(26,38)(29,42)(30,40)(33,48)(34,39)(36,47)(37,44)(41,45)(43,46);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,23)(18,24,28,27)(20,32,34,29)(21,30,38,37)(25,31,42,39)(26,40,35,44)(33,46,36,45)(41,47,43,48);(10,14)(11,16)(17,20,23,32,22,34,19,29)(18,25,27,31,28,42,24,39)(21,36,37,45,38,33,30,46)(26,43,44,48,35,41,40,47);(2,5)(6,8)(10,14)(13,15)(17,21)(18,26)(19,30)(20,33)(22,38)(23,37)(24,40)(25,41)(27,44)(28,35)(29,45)(31,47)(32,46)(34,36)(39,48)(42,43);(1,3,4,7)(2,6,5,8)(17,19,22,23)(18,27,28,24)(20,29,34,32)(21,37,38,30)(25,31,42,39)(26,40,35,44)(33,46,36,45)(41,48,43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6757304);