C₂²×#15687 #72296872

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

Aliases

direct product

C₂²×#15687

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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1268761, C₂²×#2037594, C₂²×#2310627
maximal normal subgroup of	C₂²×#1268761, C₂²×#2037594, C₂²×#2310627

GAP

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

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

C22×#15687 #72296872