C₂²×#15451 #72296883

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂²×#15451

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

52605 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#2822351, C₂²×#633924, C₂²×#3825309
maximal normal subgroup of	C₂²×#2822351, C₂²×#633924, C₂²×#3825309

GAP

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

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

C22×#15451 #72296883