C₂²×#16320 #72294972

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#16320

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

56933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

(10,12)(14,17)(18,20)(21,22,24,25)(23,27,28,26);(9,10,11,12)(14,17)(15,19)(22,25)(26,27);(6,8)(13,14,16,17)(15,18,19,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,8)(13,15)(14,18)(16,19)(17,20)(21,23,24,28)(22,26,25,27);(3,4);(1,2)(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293959, C₂×#72293967, C₂²×#4940920
maximal normal subgroup of	C₂×#72293959, C₂×#72293967, C₂²×#4940920

GAP

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

gap> G := Group((10,12)(14,17)(18,20)(21,22,24,25)(23,27,28,26), (9,10,11,12)(14,17)(15,19)(22,25)(26,27), (6,8)(13,14,16,17)(15,18,19,20)(21,22)(23,26)(24,25)(27,28), (5,6)(7,8)(13,15)(14,18)(16,19)(17,20)(21,23,24,28)(22,26,25,27), (3,4), (1,2)(3,4));

C22×#16320 #72294972