#6795621

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(17,18,24,30,23,29,22,28)(19,31,35,42,34,25,33,43)(20,27,37,45,21,38,39,26)(32,41,47,40,36,46,48,44);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,25)(20,36)(21,32)(22,33)(23,34)(24,35)(26,44)(27,41)(28,42)(29,31)(30,43)(37,48)(38,46)(39,47)(40,45);(2,5)(6,8)(10,13)(14,15)(17,20,23,21)(18,26,29,45)(19,32,34,36)(22,37,24,39)(25,40,31,44)(27,30,38,28)(33,47,35,48)(41,42,46,43);(1,3,4,7)(2,6,5,8)(17,21,23,20)(18,27,29,38)(19,32,34,36)(22,37,24,39)(25,41,31,46)(26,30,45,28)(33,48,35,47)(40,42,44,43);(9,11,12,16)(10,14,13,15)(17,22,23,24)(18,28,29,30)(19,33,34,35)(20,37,21,39)(25,42,31,43)(26,27,45,38)(32,47,36,48)(40,46,44,41)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6702817);