C₂×#3861012 #72325397

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#3861012

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

47755 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232097, C₂×#72232149, C₂×#72232126, …
maximal normal subgroup of	C₂×#72232097, C₂×#72232149, C₂×#72232126, …

GAP

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

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

C2×#3861012 #72325397