C₂²×#62957 #72299721

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#62957

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

477289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 28 points:

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

Conjugacy classes & character table

292 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#10586570
maximal normal subgroup of	C₂²×#10586570
maximal quotient of	C₂²×#7852137, C₂²×#7968016, C₂²×#7687886

GAP

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

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

C22×#62957 #72299721