C₂²×#58990 #72236635

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

Aliases

direct product

C₂²×#58990

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

298865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 16 points:

(3,4);(1,2)(3,4);(9,10)(11,12)(13,14,15,16);(6,8)(9,10,12,11)(13,14,15,16);(6,8)(10,11)(13,14)(15,16);(10,11);(5,6,7,8)(9,11,12,10)(13,14)(15,16)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7792761, C₂²×#7684106, C₂²×#7670203, …
maximal normal subgroup of	C₂²×#7792761, C₂²×#7684106, C₂²×#7670203, …
maximal quotient of	C₂²×#647656, C₂²×#648360, C₂×#72219693, …

GAP

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

gap> G := Group((3,4), (1,2)(3,4), (9,10)(11,12)(13,14,15,16), (6,8)(9,10,12,11)(13,14,15,16), (6,8)(10,11)(13,14)(15,16), (10,11), (5,6,7,8)(9,11,12,10)(13,14)(15,16));

C22×#58990 #72236635