C₂²×#14311 #72321589

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#14311

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

72149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1393451, C₂²×#1393449, C₂²×#517627, …
maximal normal subgroup of	C₂²×#1393451, C₂²×#1393449, C₂²×#517627, …
maximal quotient of	C₂²×#154635

GAP

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

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

C22×#14311 #72321589