C₂×#4932389 #72325211

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#4932389

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

51833 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232093, C₂×#72232150, C₂×#72232131
maximal normal subgroup of	C₂×#72232093, C₂×#72232150, C₂×#72232131

GAP

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

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

C2×#4932389 #72325211