C₂²×#62880 #72299720

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

Aliases

direct product

C₂²×#62880

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

478053 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)(7,12)(8,10)(9,11)(13,14)(15,20)(16,21)(17,23)(18,19)(22,28)(24,26)(25,27);(5,7)(6,8)(9,10)(11,12)(14,19)(16,25)(20,26)(22,23);(5,6)(7,12)(8,10)(9,11)(13,15,18,24)(14,20,19,26)(16,23,25,22)(17,27,28,21);(5,6)(7,8)(9,11)(10,12)(13,15)(14,20)(16,23)(17,21)(18,24)(19,26)(22,25)(27,28);(5,8)(6,10)(7,11)(9,12)(13,16,18,25)(14,21,19,27)(15,23,24,22)(17,26,28,20);(5,8)(6,10)(7,11)(9,12)(13,17)(14,22)(15,21)(16,26)(18,28)(19,23)(20,25)(24,27);(3,4);(1,2)(3,4)

Conjugacy classes & character table

280 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×#7779660, C₂²×#7779984

GAP

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

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

C22×#62880 #72299720