C₂²×#35088 #72339330

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#35088

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

312013 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7630442
maximal normal subgroup of	C₂²×#7630442
maximal quotient of	C₂×#72339318, C₂×#72334925, C₂²×#613188

GAP

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

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

C22×#35088 #72339330