C₂×#617099 #72325619

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

Aliases

direct product

C₂×#617099

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

38865 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232140, C₂×#72232138, C₂×#72232186
maximal normal subgroup of	C₂×#72232140, C₂×#72232138, C₂×#72232186

GAP

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

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

C2×#617099 #72325619