C₂×#847337 #72341778

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#847337

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

51725 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)(11,12,14,16)(13,17,18,15)(22,26)(23,25);(1,2)(11,13)(12,15)(14,18)(16,17)(19,20,24,21)(22,23,26,25);(11,13)(12,15)(14,18)(16,17)(19,21,24,20)(22,25,26,23);(12,16)(15,17)(19,22,24,26)(20,25,21,23);(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,26)(21,22)(24,25);(3,4)(5,6)(7,8)(9,10)(11,13)(12,15)(14,18)(16,17)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72327991
maximal normal subgroup of	C₂×#72327991

GAP

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

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

C2×#847337 #72341778