C₂×#696094 #72323906

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

Aliases

direct product

C₂×#696094

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

57567 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);(4,6)(8,10)(19,20,23,26)(21,24,25,22);(4,6)(7,8)(9,10)(11,12,14,16)(13,17,18,15)(19,21,23,25)(20,24,26,22);(3,4)(5,6)(11,12)(13,17)(14,16)(15,18)(19,21)(20,24)(22,26)(23,25);(1,2);(3,4)(5,6)(11,13,14,18)(12,15,16,17)(19,22,23,24)(20,25,26,21)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323252, C₂×#72323247, C₂×#72323267, …
maximal normal subgroup of	C₂×#72323252, C₂×#72323247, C₂×#72323267, …

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

C2×#696094 #72323906