C₂×#990415 #72325095

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

Aliases

direct product

C₂×#990415

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

41545 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)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20,24,26)(21,25,22,23);(3,4,6,5)(12,13)(16,17)(21,22)(23,25);(7,8)(9,10)(19,21,24,22)(20,25,26,23);(4,5)(7,8,9,10)(11,13,15,12)(14,16,18,17)(19,22,24,21)(20,23,26,25);(1,2);(3,5,6,4)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,23)(20,22)(21,26)(24,25)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232183, C₂×#72232165, C₂×#72232164
maximal normal subgroup of	C₂×#72232183, C₂×#72232165, C₂×#72232164

GAP

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

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

C2×#990415 #72325095