C₂²×#59267 #72296607

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

Aliases

direct product

C₂²×#59267

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

89909 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:

(13,14)(15,19)(16,17)(18,20)(21,22)(23,27)(24,28)(25,26);(21,23)(22,24)(25,28)(26,27);(6,9)(10,12)(14,17)(15,20)(22,26)(24,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,26)(27,28);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,18,17,19)(21,24)(22,23)(25,27)(26,28);(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×#72232346, C₂×#72232287
maximal normal subgroup of	C₂²×#8181078, C₂×#72232346, C₂×#72232287
maximal quotient of	C₂²×#4030235, C₂²×#1427668

GAP

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

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

C22×#59267 #72296607