C₂²×#49563 #72324876

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

Aliases

direct product

C₂²×#49563

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

99129 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7779659
maximal normal subgroup of	C₂²×#7779659

GAP

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

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

C22×#49563 #72324876