#6800813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 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)(17,18,23,29)(19,30,33,25)(20,27,21,37)(22,28,24,39)(26,38,44,36)(31,41,35,45)(32,46,34,42)(40,47,43,48);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,19)(18,25)(20,35)(21,31)(22,32)(23,33)(24,34)(26,43)(27,41)(28,42)(29,30)(36,48)(37,45)(38,47)(39,46)(40,44);(2,5)(6,8)(10,13)(14,15)(17,20,23,21)(18,26,29,44)(19,31,33,35)(22,36,24,38)(25,40,30,43)(27,28,37,39)(32,47,34,48)(41,46,45,42);(1,3,4,7)(2,6,5,8)(17,21,23,20)(18,27,29,37)(19,31,33,35)(22,36,24,38)(25,41,30,45)(26,28,44,39)(32,48,34,47)(40,46,43,42);(9,11,12,16)(10,14,13,15)(17,22,23,24)(18,28,29,39)(19,32,33,34)(20,36,21,38)(25,42,30,46)(26,37,44,27)(31,47,35,48)(40,41,43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708009);