#6460251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

1546 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)(19,28)(20,26)(21,27)(22,24)(23,36)(25,32)(29,43)(30,42)(31,37)(33,38)(34,40)(35,41)(39,45)(44,48)(46,47);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,24)(19,29)(20,33)(27,36)(31,32)(34,35)(39,47)(42,48);(10,14)(11,15)(17,19,21,31,22,29,23,32)(18,25,36,43,24,37,27,28)(20,30,35,46,33,44,34,45)(26,39,40,48,38,47,41,42);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,33)(18,26,24,38)(19,30,29,44)(21,35,23,34)(25,39,37,47)(27,41,36,40)(28,42,43,48)(31,46,32,45);(17,21,22,23)(18,27,24,36)(19,31,29,32)(20,34,33,35)(25,28,37,43)(26,40,38,41)(30,45,44,46)(39,48,47,42)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367447);