C₂²×#53167 #72325898

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

Aliases

direct product

C₂²×#53167

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

67953 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)(7,11,12,9)(15,19)(18,20)(21,22,24,27)(23,26,28,25);(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,17)(19,20)(21,23,24,28)(22,25,27,26);(7,12)(9,11)(14,17)(15,19)(21,23)(22,26)(24,28)(25,27);(3,4);(5,6)(7,11)(8,10)(9,12)(13,15,16,19)(14,18,17,20)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4);(5,7)(6,9)(8,12)(10,11)(13,14)(15,20)(16,17)(18,19)(21,22)(23,26)(24,27)(25,28)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8108129
maximal normal subgroup of	C₂²×#8108129

GAP

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

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

C22×#53167 #72325898