#333406

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,13,11,9,8,17)(7,14,15,18,16,20,19,12)(22,27)(23,31)(29,30)(33,34);(5,7,11,16)(6,12,9,18)(8,15,10,19)(13,20,17,14)(21,22,25,27)(23,32,31,36)(24,30,26,29)(28,34,35,33);(2,4)(5,8,11,10)(6,13,9,17)(7,15,16,19)(12,20,18,14)(21,23,25,31)(22,28,27,35)(24,33,26,34)(29,32,30,36);(2,4)(5,9,11,6)(7,14,16,20)(8,17,10,13)(12,19,18,15)(21,24,25,26)(22,29,27,30)(23,33,31,34)(28,32,35,36)

Polycyclic presentation:

2^9|a^2=ei,b^2=hi,c^2=hi,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240602);