#322355

order	512
nilpotency class	3
isoclinism	isoclinic class of #322325
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1712	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 162 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)(3,10)(4,6)(5,9)(7,12)(8,11)(13,14)(15,16)(17,18,24,27,23,20,22,28)(19,26,21,31,30,32,29,25);(2,6,9,8)(7,14,15,10)(17,19,22,29,23,30,24,21)(18,25,27,26,20,31,28,32);(1,2)(3,7)(4,8)(5,9)(6,11)(10,12)(13,15)(14,16)(17,20,23,18)(19,25,30,31)(21,32,29,26)(22,28,24,27);(1,3,11,16,5,13,4,12)(2,7,6,14,9,15,8,10)(17,21,24,30,23,29,22,19)(18,26,28,25,20,32,27,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229551);