#6966709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873905);