#322351

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	C₂³	#550
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,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, 229547);