C₂²×#60032 #72295855

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

Aliases

direct product

C₂²×#60032

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

72625 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8075171
maximal normal subgroup of	C₂²×#8075171

GAP

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

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

C22×#60032 #72295855