C₂×#1082837 #72325269

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

Aliases

direct product

C₂×#1082837

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

40007 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232187, C₂×#72232116, C₂×#72232103
maximal normal subgroup of	C₂×#72232187, C₂×#72232116, C₂×#72232103

GAP

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

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

C2×#1082837 #72325269