C₂²×#56291 #72325953

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂²×#56291

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

89829 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8387575
maximal normal subgroup of	C₂²×#8387575
maximal quotient of	C₂²×#2110333

GAP

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

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

C22×#56291 #72325953