C₂×#1393359 #72302563

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

Aliases

direct product

C₂×#1393359

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

29103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279893, C₂×#72280420, C₂×#72279836
maximal normal subgroup of	C₂×#72279893, C₂×#72280420, C₂×#72279836

GAP

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

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

C2×#1393359 #72302563