#327315

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35165	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1372 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1372 conjugacy classes of subgroups — too many to draw. Only its 152 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)(10,13)(11,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,21,26)(18,22,24,29)(20,27,28,25)(23,31,32,30);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(22,29)(30,31);(3,7)(6,8)(11,15)(14,16)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,30,29,31)

Polycyclic presentation:

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

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, 234511);