C₂×#1393214 #72325449

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

Aliases

direct product

C₂×#1393214

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

41165 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232184, C₂×#72232137, C₂×#72232143
maximal normal subgroup of	C₂×#72232184, C₂×#72232137, C₂×#72232143

GAP

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

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

C2×#1393214 #72325449