C₂×#637549 #72325245

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#637549

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

49677 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)(8,10)(12,15)(13,17)(19,20,22,25)(21,23,26,24);(4,5)(8,10)(13,17)(16,18)(19,21,22,26)(20,23,25,24);(21,26)(23,24);(7,8)(9,10)(11,12)(13,18)(14,15)(16,17)(19,20)(21,24)(22,25)(23,26);(1,2);(3,5,6,4)(11,13,14,17)(12,16,15,18)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232082, C₂×#72232108, C₂×#72232147
maximal normal subgroup of	C₂×#72232082, C₂×#72232108, C₂×#72232147

GAP

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

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

C2×#637549 #72325245