C₂²×#56346 #72291781

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂²×#56346

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

90673 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)(7,11)(14,15)(16,20)(22,25)(26,28);(7,11)(10,12)(13,14,17,15)(16,19,20,18)(21,22,24,25)(23,26,27,28);(14,15)(18,19)(23,27)(26,28);(5,6,8,9)(7,12,11,10)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,25)(27,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8676694, C₂×#72278344, C₂²×#9164052, …
maximal normal subgroup of	C₂²×#8676694, C₂×#72278344, C₂²×#9164052, …
maximal quotient of	C₂²×#2805719, C₂²×#3868090, C₂²×#4029237, …

GAP

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

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

C22×#56346 #72291781