#6765389

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6672585);