C₂×#946416 #72324985

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

Aliases

direct product

C₂×#946416

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

40603 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232172, C₂×#72232125, C₂×#72232105
maximal normal subgroup of	C₂×#72232172, C₂×#72232125, C₂×#72232105

GAP

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

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

C2×#946416 #72324985