#6539635

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1360 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)(3,7)(4,8)(5,6)(10,11)(14,15)(17,18,23,26)(19,24,31,38)(20,32,34,42)(21,28,22,27)(25,36,40,33)(29,43,46,48)(30,41,35,39)(37,44,47,45);(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,31)(18,24,26,38)(20,29,34,46)(21,35,22,30)(25,37,40,47)(27,41,28,39)(32,43,42,48)(33,45,36,44);(1,3,5,4)(2,7,6,8)(9,11,13,10)(12,15,16,14)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,40)(27,32)(28,42)(30,44)(31,46)(35,45)(38,47)(39,48)(41,43);(1,4,5,3)(2,8,6,7)(12,16)(14,15)(18,26)(24,38)(25,40)(27,28)(32,42)(37,47)(39,41)(43,48);(9,12,13,16)(10,14,11,15)(17,21,23,22)(18,27,26,28)(19,30,31,35)(20,33,34,36)(24,39,38,41)(25,32,40,42)(29,44,46,45)(37,48,47,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6446831);