C₂²×#50011 #72296211

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

Aliases

direct product

C₂²×#50011

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

79085 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#9228811
maximal normal subgroup of	C₂²×#9228811
maximal quotient of	C₂²×#3860970

GAP

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

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

C22×#50011 #72296211