C₂×#515421 #72339344

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#515421

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

72189 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237462, C₂×#72235253, C₂×#72237463, …
maximal normal subgroup of	C₂×#72237462, C₂×#72235253, C₂×#72237463, …
maximal quotient of	C₂×#72323614, #72340070

GAP

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

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

C2×#515421 #72339344