#6457908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365104);