C₂²×#35057 #72337191

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#35057

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

113181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7628282, C₂²×#7642257
maximal normal subgroup of	C₂²×#7628282, C₂²×#7642257

GAP

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

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

C22×#35057 #72337191