#324475

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

572 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231671);