#6840642

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 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)(9,10)(11,14)(12,13)(15,16);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,22,26)(19,27,31,42)(20,32,33,46)(21,35,28,25)(23,36,38,30)(24,39,40,34)(29,43,44,37)(41,47,48,45);(10,13)(14,16)(17,19,28,36,22,31,21,30)(18,23,35,42,26,38,25,27)(20,29,39,47,33,44,34,45)(24,37,46,48,40,43,32,41);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(20,33)(24,40)(29,44)(32,46)(34,39)(37,43)(41,48)(45,47);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,34,22,33,28,39)(18,24,25,32,26,40,35,46)(19,29,30,45,31,44,36,47)(23,37,27,41,38,43,42,48)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=fg,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6747838);