C₂²×#52397 #72326142

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

Aliases

direct product

C₂²×#52397

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

90749 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8598997
maximal normal subgroup of	C₂²×#8598997
maximal quotient of	C₂²×#1006038

GAP

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

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

C22×#52397 #72326142