C₂×#4308918 #72339345

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄³

Aliases

direct product

C₂×#4308918

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

82395 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)(7,8)(9,10)(11,12,14,16)(13,17,18,15)(21,25)(23,26);(8,10)(13,18)(15,17)(19,20)(21,23)(22,24)(25,26);(4,5)(8,10)(19,20,22,24)(21,23,25,26);(3,5,6,4)(7,8)(9,10)(11,13)(12,15)(14,18)(16,17)(19,20)(21,26)(22,24)(23,25);(1,2);(11,12)(13,17)(14,16)(15,18)(19,21)(20,23)(22,25)(24,26)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237451, C₂×#72237465
maximal normal subgroup of	C₂×#72237451, C₂×#72237465
maximal quotient of	C₂×#72339321

GAP

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

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

C2×#4308918 #72339345