C₂×#5955399 #72339415

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

Aliases

direct product

C₂×#5955399

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

44457 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72327589, C₂×#72327592, C₂×#72327623
maximal normal subgroup of	C₂×#72327589, C₂×#72327592, C₂×#72327623

GAP

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

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

C2×#5955399 #72339415