#6826911

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#24265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(9,10,13,15)(11,16,12,14)(17,18)(19,25)(20,26)(21,27)(22,37)(23,29)(24,30)(28,38)(31,39)(32,40)(33,41)(34,42)(35,44)(36,45)(43,47)(46,48);(9,10)(11,14)(12,16)(13,15)(17,19,24,32)(18,25,30,40)(20,33,36,47)(21,22,23,38)(26,41,45,43)(27,37,29,28)(31,34,46,35)(39,42,48,44);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,23,35,24,36,21,34)(18,26,29,44,30,45,27,42)(19,31,22,33,32,46,38,47)(25,39,37,41,40,48,28,43);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(17,21,24,23)(18,27,30,29)(19,22,32,38)(20,34,36,35)(25,37,40,28)(26,42,45,44)(31,33,46,47)(39,41,48,43);(9,12,13,11)(10,14,15,16)(17,22,23,32,24,38,21,19)(18,28,29,25,30,37,27,40)(20,33,35,46,36,47,34,31)(26,43,44,39,45,41,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fh,d^2=gi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6734107);