C₂²×#59864 #72301382

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#59864

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#127	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

62369 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 28 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7985294, C₂²×#7667013, C₂³×#59864
maximal normal subgroup of	C₂²×#7985294, C₂²×#7667013, C₂³×#59864
maximal quotient of	C₂³×#59864

GAP

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

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

C22×#59864 #72301382