#411278

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1042 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=gh,c^2=fgi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318474);