C₂²×#48563 #72296690

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

Aliases

direct product

C₂²×#48563

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

79929 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72230859, C₂²×#8757652
maximal normal subgroup of	C₂×#72230859, C₂²×#8757652
maximal quotient of	C₂²×#3868090

GAP

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

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

C22×#48563 #72296690