#451127

order	512
nilpotency class	3
isoclinism	isoclinic class of #417496
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1750 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1750 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358323);