C₂×#3858151 #72296313

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

Aliases

direct product

C₂×#3858151

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

38033 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)(13,17)(16,18);(3,4,5,6)(7,8)(9,10)(11,12,14,15)(13,18,17,16)(19,20)(21,22)(23,26)(24,25);(3,4,5,6)(12,15)(13,17)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,25)(24,26);(3,4)(5,6)(11,13)(12,16)(14,17)(15,18)(19,22)(20,25)(21,26)(23,24);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278043
maximal normal subgroup of	C₂×#72278043
maximal quotient of	C₂×#72278148

GAP

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

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

C2×#3858151 #72296313