C₂×#618321 #72324430

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#618321

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

51953 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)(11,12)(13,15)(14,16)(17,18)(19,20,23,22)(21,25,26,24);(4,6)(11,13,14,18)(12,15,16,17)(19,20,23,22)(21,25,26,24);(8,10)(12,16)(13,18)(19,21,23,26)(20,24,22,25);(11,13)(12,17)(14,18)(15,16)(19,21)(20,25)(22,24)(23,26);(1,2);(3,4,5,6)(7,8,9,10)(11,12)(13,17)(14,16)(15,18)(19,22,23,20)(21,25,26,24)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232038, C₂×#72232041, C₂×#72232055
maximal normal subgroup of	C₂×#72232038, C₂×#72232041, C₂×#72232055

GAP

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

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

C2×#618321 #72324430