#6435919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433042
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 538 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343115);