#333401

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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 780 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)(5,6,10,11,9,14,7,12)(8,16,15,13,18,19,17,20)(21,22)(23,31)(24,29)(25,27)(26,30)(28,33)(32,35)(34,36);(6,11)(7,10)(8,17)(12,14)(13,20)(15,18)(22,27)(23,32)(24,26)(28,35)(31,36)(33,34);(2,4)(5,7,9,10)(6,12,14,11)(8,15,18,17)(13,19,20,16)(21,23,25,33)(22,28,27,31)(24,32,26,34)(29,36,30,35);(2,4)(5,8,9,18)(6,13,14,20)(7,15,10,17)(11,16,12,19)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,36,31,35)

Polycyclic presentation:

2^9|a^2=ehi,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, 240597);