C₂×#686404 #72332830

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#686404

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

30157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323393, C₂×#72323305, C₂×#72323401
maximal normal subgroup of	C₂×#72323393, C₂×#72323305, C₂×#72323401

GAP

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

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

C2×#686404 #72332830