C₂×#4453257 #72324433

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

Aliases

direct product

C₂×#4453257

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

73657 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)(8,10)(12,15)(16,18)(19,20,22,23)(21,24,25,26);(13,17)(16,18)(20,23)(21,25);(21,25)(24,26);(8,10)(11,12,14,15)(13,16,17,18)(19,21)(20,24)(22,25)(23,26);(1,2);(3,4,5,6)(7,8,9,10)(11,13)(12,16)(14,17)(15,18)(19,20)(21,26)(22,23)(24,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232060, C₂×#72232053, C₂×#72232040
maximal normal subgroup of	C₂×#72232060, C₂×#72232053, C₂×#72232040

GAP

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

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

C2×#4453257 #72324433