#213930

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36707	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1038 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121126);