C₂²×#16508 #72291892

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#16508

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

102037 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#2920917, C₂²×#3862142, C₂²×#719145, …
maximal normal subgroup of	C₂²×#2920917, C₂²×#3862142, C₂²×#719145, …
maximal quotient of	C₂²×#156706, C₂³×#16508, C₂×#72334411

GAP

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

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

C22×#16508 #72291892