C₂²×#15091 #72339133

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂²×#15091

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

83277 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#618316, C₂²×#618327, C₂²×#608727, …
maximal normal subgroup of	C₂²×#618316, C₂²×#618327, C₂²×#608727, …
maximal quotient of	C₂×#72335061

GAP

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

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

C22×#15091 #72339133