C₂²×#45599 #72291789

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#45599

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

73697 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)(18,19)(21,22)(23,27)(24,26)(25,28);(21,22,24,26)(23,25,28,27);(7,11)(10,12)(13,14,17,15)(16,18,20,19)(21,22)(23,27)(24,26)(25,28);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,18,20,19)(21,22)(23,27)(24,26)(25,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,25)(24,28)(26,27);(3,4);(1,2)(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8676694, C₂×#72293910, C₂²×#7985294, …
maximal normal subgroup of	C₂²×#8676694, C₂×#72293910, C₂²×#7985294, …
maximal quotient of	C₂²×#845354, C₂²×#5021210, C₂³×#45599

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

C22×#45599 #72291789