#6742018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#24124	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1577 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)(9,10,12,14)(11,15,16,13)(18,23)(19,27)(20,31)(26,35)(30,36)(33,45)(39,41)(43,46);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,23)(19,28,27,40)(20,25,31,38)(22,35,34,26)(24,36,37,30)(29,43,32,46)(33,48,45,42)(39,47,41,44);(10,14)(13,15)(17,19,21,27)(18,24,23,37)(20,32,31,29)(22,36,34,30)(25,41,38,39)(26,40,35,28)(33,44,45,47)(42,43,48,46);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,20,21,31)(18,25,23,38)(19,29,27,32)(22,33,34,45)(24,39,37,41)(26,42,35,48)(28,43,40,46)(30,44,36,47);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(18,26)(22,34)(23,35)(24,40)(25,42)(28,37)(30,36)(33,45)(38,48)(39,46)(41,43)(44,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6649214);