C₂²×#1356 #483007

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×#1356

Subgroups

	subgroup	G/H
center, socle	C₂³	#329
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

822 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 242 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);(5,6,8,13,9,14,22,32,10,15,21,33,24,12,29,30)(7,16,17,34,18,11,25,26,19,27,35,31,23,28,20,36);(5,7,10,19)(6,11,15,28)(8,20,21,25)(9,23,24,18)(12,27,14,16)(13,31,33,34)(17,22,35,29)(26,30,36,32)

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, 390203);