#6767409

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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)(19,25)(20,35)(21,27)(22,28)(23,38)(24,29)(26,36)(30,45)(31,40)(32,41)(33,48)(34,42)(37,44)(39,46)(43,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,24,34)(18,25,29,42)(20,31,21,30)(22,32,23,33)(26,43,44,39)(27,45,35,40)(28,41,38,48)(36,47,37,46);(10,13)(14,16)(17,20,24,21)(18,26,29,44)(19,30,34,31)(22,36,23,37)(25,39,42,43)(27,38,35,28)(32,46,33,47)(40,48,45,41);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,29,35)(19,31,34,30)(22,36,23,37)(25,40,42,45)(26,38,44,28)(32,46,33,47)(39,48,43,41);(3,8)(5,7)(10,13)(14,16)(17,22,24,23)(18,28,29,38)(19,32,34,33)(20,36,21,37)(25,41,42,48)(26,27,44,35)(30,46,31,47)(39,40,43,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6674605);