C₂×#2108860 #72325006

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#2108860

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

48105 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)(13,15)(14,16)(17,18)(19,20)(21,24)(22,25)(23,26);(3,4,5,6)(11,12,14,16)(13,15,18,17)(19,21)(20,22)(23,25)(24,26);(8,10)(19,21,23,25)(20,24,26,22);(4,6)(7,8)(9,10)(11,13,14,18)(12,15,16,17)(19,21)(20,22)(23,25)(24,26);(1,2);(3,4)(5,6)(11,12)(13,17)(14,16)(15,18)(19,22)(20,25)(21,26)(23,24)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232168, C₂×#72232137, C₂×#72232153
maximal normal subgroup of	C₂×#72232168, C₂×#72232137, C₂×#72232153

GAP

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

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

C2×#2108860 #72325006