C₂²×#59370 #72296610

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

Aliases

direct product

C₂²×#59370

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

89065 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)(7,11)(8,9)(10,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,27)(24,26)(25,28);(13,15)(14,17)(16,20)(18,19);(6,9)(10,11)(21,22,24,26)(23,25,28,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,23)(22,25)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,19)(18,20)(21,22)(23,27)(24,26)(25,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8181078, C₂×#72232339
maximal normal subgroup of	C₂²×#8181078, C₂×#72232339
maximal quotient of	C₂²×#1427668, C₂²×#850480

GAP

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

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

C22×#59370 #72296610