C₂²×#11598 #72287434

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#11598

Subgroups

	subgroup	G/H
center	C₂³×C₄	#102
commutator	C₈	C₂⁴×C₈
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂⁴	C₂×#102
prime radical	C₂³×#135	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

7797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#483590, C₂²×#484053, C₂²×#481415
maximal normal subgroup of	C₂²×#483590, C₂²×#484053, C₂²×#481415
maximal quotient of	C₂²×#136613

GAP

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

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

C22×#11598 #72287434