C₂²×#57965 #72301271

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂²×#57965

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

70053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8757652
maximal normal subgroup of	C₂²×#8757652

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

C22×#57965 #72301271