#326830

order	512
nilpotency class	3
isoclinism	isoclinic class of #326783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1022 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,6)(4,5)(7,8)(9,10,13,11)(12,16,14,15)(17,18)(19,26)(20,30)(21,25)(22,24)(23,27)(28,31)(29,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26)(20,25)(21,30)(22,24)(23,28)(27,31)(29,32);(1,3)(2,6)(4,7)(5,8)(10,15)(11,16)(12,14)(17,19,22,29)(18,23,24,31)(20,28,21,27)(25,32,30,26);(3,7)(6,8)(9,12,13,14)(10,15,11,16)(18,24)(23,31)(25,30)(26,32)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gh,d^c=de,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234026);