#322334

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	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 672 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=egh,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, 229530);