C₂×#991661 #72325437

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

Aliases

direct product

C₂×#991661

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

54997 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)(11,12,15,13)(14,17,18,16)(19,20,23,24)(21,22,25,26);(3,4,5,6)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,25)(24,26);(7,8)(9,10)(12,13)(16,17)(20,24)(21,25);(4,6)(7,8,9,10)(11,13,15,12)(14,17,18,16)(19,21)(20,22)(23,25)(24,26);(1,2);(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,26)(24,25)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232087, C₂×#72232153, C₂×#72232134
maximal normal subgroup of	C₂×#72232087, C₂×#72232153, C₂×#72232134

GAP

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

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

C2×#991661 #72325437