C₂×#1393222 #72325561

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

Aliases

direct product

C₂×#1393222

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

40653 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232189, C₂×#72232184, C₂×#72232185
maximal normal subgroup of	C₂×#72232189, C₂×#72232184, C₂×#72232185

GAP

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

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

C2×#1393222 #72325561