#6458759

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6365955);