#6822727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1320 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)(10,14)(11,12)(17,18,21,25)(19,27,29,23)(20,31,34,24)(22,26,35,40)(28,41,42,36)(30,37,33,39)(32,45,44,38)(43,47,46,48);(9,10)(11,15)(12,16)(13,14)(19,28)(20,32)(22,35)(23,36)(24,38)(26,40)(27,41)(29,42)(30,33)(31,45)(34,44)(37,39);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19,22,28,21,29,35,42)(18,23,26,36,25,27,40,41)(20,33,32,46,34,30,44,43)(24,39,38,48,31,37,45,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(19,29)(23,27)(28,42)(30,33)(36,41)(37,39)(43,46)(47,48);(10,14)(15,16)(17,20,22,32,21,34,35,44)(18,24,26,38,25,31,40,45)(19,30,28,43,29,33,42,46)(23,37,36,47,27,39,41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=gi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6729923);