#6911376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1731 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6818572);