C₂×#515557 #72331088

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#515557

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#534	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

75917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323284, C₂×#72323286, C₂×#72323283
maximal normal subgroup of	C₂×#72323284, C₂×#72323286, C₂×#72323283

GAP

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

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

C2×#515557 #72331088