C₂²×#14825 #72293283

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂²×#14825

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

76085 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1420724, C₂²×#1260433, C₂²×#4929019, …
maximal normal subgroup of	C₂²×#1420724, C₂²×#1260433, C₂²×#4929019, …
maximal quotient of	C₂³×#14825

GAP

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

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

C22×#14825 #72293283