C₂²×#1288 #181208

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂²×#1288

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#9428, C₂²×#10018, C₂²×#9492
maximal normal subgroup of	C₂²×#9428, C₂²×#10018, C₂²×#9492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88404);

C22×#1288 #181208

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂²×#1288 #181208