C₂×#990414 #72325384

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

Aliases

direct product

C₂×#990414

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

41513 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232179, C₂×#72232135, C₂×#72232105
maximal normal subgroup of	C₂×#72232179, C₂×#72232135, C₂×#72232105

GAP

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

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

C2×#990414 #72325384