C₂×#523321 #72332856

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#523321

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

51921 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323393, C₂×#72323356, C₂×#72323401
maximal normal subgroup of	C₂×#72323393, C₂×#72323356, C₂×#72323401

GAP

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

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

C2×#523321 #72332856