#451126

order	512
nilpotency class	3
isoclinism	isoclinic class of #417496
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2550 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 2550 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(2,6)(5,8)(11,15)(13,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(20,28)(22,25)(23,30)(26,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^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, 358322);