C₂²×#1340 #481436

order	512
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1340

Subgroups

	subgroup	G/H
center, socle	C₂³	#327
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1238 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1238 conjugacy classes of subgroups — too many to draw. Only its 282 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) = 128

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388632);