C₂×#2125677 #72325192

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#2125677

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

46165 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)(9,10)(11,12,15,14)(13,16,17,18)(19,20)(21,22)(23,25)(24,26);(3,4)(5,6)(7,8)(9,10)(11,12,15,14)(13,16,17,18)(19,21)(20,24)(22,25)(23,26);(7,8)(9,10)(12,14)(16,18);(4,5)(8,10)(11,13,15,17)(12,16,14,18)(19,22,23,24)(20,21,25,26);(1,2);(3,5,6,4)(7,8)(9,10)(11,14,15,12)(13,16,17,18)(19,21)(20,22)(23,26)(24,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232091, C₂×#72232173, C₂×#72232095, …
maximal normal subgroup of	C₂×#72232091, C₂×#72232173, C₂×#72232095, …

GAP

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

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

C2×#2125677 #72325192