C₂×#1297985 #72331928

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#1297985

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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323295, C₂×#72323314, C₂×#72300631
maximal normal subgroup of	C₂×#72323295, C₂×#72323314, C₂×#72300631

GAP

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

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

C2×#1297985 #72331928