C₂²×#45271 #72299752

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

Aliases

direct product

C₂²×#45271

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

89613 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)(12,16)(14,15)(19,23)(21,24);(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,21,23,24);(9,10,13,11)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(9,11,13,10)(12,14,16,15)(17,19,20,23)(18,21,22,24);(9,12,13,16)(10,14,11,15)(17,18)(19,24)(20,22)(21,23);(3,4);(1,2)(3,4)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7758045
maximal normal subgroup of	C₂²×#7758045

GAP

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

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

C22×#45271 #72299752