#496395

order	512
nilpotency class	4
isoclinism	isoclinic class of #480674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1180 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 210 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,25,29,23)(22,27)(28,31,32,30);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,31,25,30);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(19,27)(21,29)(22,30)(24,25)(26,31);(2,8)(6,9)(7,16)(14,15)(18,24)(22,26)(23,25)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fi,g^c=ghi,g^f=gi

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, 403591);