#326800

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	#35001	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1510 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1510 conjugacy classes of subgroups — too many to draw. Only its 160 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)(11,12)(13,16)(14,15)(17,18)(19,24)(20,30)(21,27)(22,26)(23,28)(25,31)(29,32);(2,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,28)(21,29)(22,25)(23,30)(26,31)(27,32);(1,3)(2,6)(4,7)(5,8)(10,11)(12,14)(15,16)(17,20,23,24)(18,25,28,29)(19,26,30,27)(21,31,22,32);(3,7)(6,8)(9,12,13,14)(10,15,16,11)(17,21,23,22)(18,26,28,27)(19,25,30,29)(20,31,24,32)

Polycyclic presentation:

2^9|c^2=hi,d^2=h,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, 233996);