C₂²×#9089 #72308485

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#9089

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₂×C₄	C₂⁴×C₈
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	C₂⁴×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

12081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#196123
maximal normal subgroup of	C₂²×#196123
maximal quotient of	#72309250

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C22×#9089 #72308485