C₂²×#59229 #72301161

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#59229

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

81117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7687886, C₂²×#7677129, C₂²×#7985575
maximal normal subgroup of	C₂²×#7687886, C₂²×#7677129, C₂²×#7985575

GAP

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

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

C22×#59229 #72301161