#6788229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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)(17,18,21,23)(19,27,28,43)(20,32,34,25)(22,36,29,26)(24,38,37,31)(30,45,33,42)(35,41,44,47)(39,46,40,48);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(18,23)(19,28)(25,32)(26,36)(30,33)(31,38)(41,47)(46,48);(2,5)(6,8)(10,13)(14,16)(17,19,29,38,21,28,22,31)(18,24,36,43,23,37,26,27)(20,33,44,46,34,30,35,48)(25,40,47,42,32,39,41,45);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,34)(18,25,23,32)(19,30,28,33)(22,35,29,44)(24,39,37,40)(26,41,36,47)(27,42,43,45)(31,46,38,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(20,34)(25,32)(30,33)(35,44)(39,40)(41,47)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6695425);