#6800759

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(17,18,21,25)(19,27,29,23)(20,31,32,40)(22,35,34,26)(24,36,39,33)(28,43,44,48)(30,38,46,42)(37,45,41,47);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,37)(25,27)(26,38)(31,48)(32,44)(33,45)(34,46)(35,42)(36,47)(39,41)(40,43);(2,5)(6,8)(10,13)(14,15)(17,20,21,32)(18,24,25,39)(19,28,29,44)(22,36,34,33)(23,37,27,41)(26,40,35,31)(30,47,46,45)(38,43,42,48);(1,3,4,7)(2,6,5,8)(18,25)(19,29)(24,39)(26,35)(28,44)(30,46)(31,40)(45,47);(9,11,12,16)(10,14,13,15)(19,29)(23,27)(28,44)(30,46)(37,41)(38,42)(43,48)(45,47)

Polycyclic presentation:

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

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, 6707955);