C₂²×#14438 #72339456

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#14438

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

64493 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1294097, C₂²×#1420611, C₂²×#525856
maximal normal subgroup of	C₂²×#1294097, C₂²×#1420611, C₂²×#525856

GAP

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

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

C22×#14438 #72339456