#6826768

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	#30397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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)(19,29)(20,33)(21,27)(22,26)(23,28)(24,31)(25,36)(30,39)(32,44)(34,40)(35,41)(37,48)(38,42)(43,46)(45,47);(9,10)(11,14)(12,16)(13,15)(17,19,22,31)(18,24,26,29)(20,34,36,38)(21,35,32,23)(25,40,33,42)(27,28,44,41)(30,46,47,37)(39,48,45,43);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,35,46,22,36,23,37)(18,25,41,48,26,33,28,43)(19,30,21,38,31,47,32,34)(24,39,44,42,29,45,27,40);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(18,26)(20,36)(24,29)(27,44)(28,41)(30,47)(34,38)(37,46);(9,12,13,11)(10,14,15,16)(17,21,35,31,22,32,23,19)(18,27,41,24,26,44,28,29)(20,34,46,30,36,38,37,47)(25,42,48,45,33,40,43,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fh,d^2=gi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6733964);