C₂×#5955371 #72325616

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#5955371

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

54449 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)(13,18)(14,16)(15,17)(22,26)(24,25);(11,13)(12,16)(14,15)(17,18);(4,6)(8,10)(12,17)(13,14)(19,20,23,21)(22,25,26,24);(1,2);(7,8,9,10)(11,14,15,13)(12,16,17,18)(19,21,23,20)(22,24,26,25);(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,18)(15,17)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232098, C₂×#72232096, C₂×#72232186, …
maximal normal subgroup of	C₂×#72232098, C₂×#72232096, C₂×#72232186, …

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

C2×#5955371 #72325616