C₂×#618312 #72342114

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#618312

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

66221 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)(8,10)(13,17)(16,18)(22,26)(24,25);(1,2)(7,8)(9,10)(12,15)(16,18)(19,20,23,21)(22,24,26,25);(7,8)(9,10)(12,15)(16,18)(19,21,23,20)(22,25,26,24);(11,12)(13,16)(14,15)(17,18)(19,21)(20,23)(22,24)(25,26);(3,4)(5,6)(7,8)(9,10)(19,22)(20,24)(21,25)(23,26);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72310477
maximal normal subgroup of	C₂×#72310477

GAP

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

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

C2×#618312 #72342114