#6936968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 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)(3,6)(4,5)(7,8)(17,18,23,27,22,26,21,28)(19,29,32,39,31,24,33,41)(20,34,36,25,35,42,37,40)(30,44,47,43,46,48,45,38);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,26,29)(20,30,35,46)(21,33,23,32)(25,38,40,43)(27,41,28,39)(34,48,42,44)(36,47,37,45);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,40)(27,34)(28,42)(29,43)(31,46)(32,45)(33,47)(39,44)(41,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,26)(19,31)(20,35)(27,28)(32,33)(36,37)(38,43)(44,48);(10,13)(14,16)(17,21,22,23)(18,27,26,28)(19,32,31,33)(20,36,35,37)(24,39,29,41)(25,34,40,42)(30,45,46,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844164);