C₂×#2033471 #72297454

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

Aliases

direct product

C₂×#2033471

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

39177 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)(11,12)(13,16)(14,17)(15,18)(19,20)(21,25)(22,24)(23,26);(4,6)(7,8,9,10)(11,13,15,17)(12,16,18,14)(19,20,22,24)(21,25,26,23);(8,10)(21,26)(23,25);(7,8)(9,10)(11,13)(12,14)(15,17)(16,18)(19,21)(20,23)(22,26)(24,25);(3,4,5,6)(11,14)(12,17)(13,18)(15,16)(19,20)(21,25)(22,24)(23,26);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72277929
maximal normal subgroup of	C₂×#72277929
maximal quotient of	C₂×#72277972

GAP

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

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

C2×#2033471 #72297454