C₂×#617132 #72325628

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

Aliases

direct product

C₂×#617132

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

38603 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)(7,8)(9,10)(12,15)(16,17)(19,20,22,24)(21,23,26,25);(7,8)(9,10)(11,12,14,15)(13,16,18,17);(8,10)(19,21)(20,23)(22,26)(24,25);(1,2);(4,6)(11,12)(13,17)(14,15)(16,18)(19,21)(20,23)(22,26)(24,25);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,20)(21,25)(22,24)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232189, C₂×#72232183, C₂×#72232186
maximal normal subgroup of	C₂×#72232189, C₂×#72232183, C₂×#72232186

GAP

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

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

C2×#617132 #72325628