C₂²×#35681 #72278751

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

Aliases

direct product

C₂²×#35681

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

117185 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)(11,13)(12,14)(15,16)(17,18)(19,20)(21,24)(22,23);(6,8)(9,10,12,14)(11,15,16,13)(17,19,21,23)(18,22,24,20);(11,16)(13,15)(17,19,21,23)(18,20,24,22);(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,24)(21,22);(3,4);(1,2)(3,4)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7752089, C₂²×#7744685
maximal normal subgroup of	C₂²×#7752089, C₂²×#7744685
maximal quotient of	C₂²×#4306358, C₂²×#778474, C₂²×#517580, …

GAP

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

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

C22×#35681 #72278751