#6916324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1387 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823520);