C₂²×#1308 #181404

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1308

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#95
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1478 conjugacy classes of subgroups; only the 618 normal subgroups are shown.

normal subgroups only

This group has 1478 conjugacy classes of subgroups — too many to draw. Only its 618 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);(5,6,8,13,10,12,9,14)(7,15,16,20,17,11,18,19)(21,22,24,27,25,28,32,35)(23,29,31,36,30,26,33,34);(5,7,9,16,10,17,8,18)(6,11,14,19,12,15,13,20)(21,23)(22,26,28,29)(24,31)(25,30)(27,34,35,36)(32,33)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88600);