C₂×#5955311 #72339397

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

Aliases

direct product

C₂×#5955311

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48037 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)(11,12)(13,16)(14,17)(15,18)(21,25)(23,26);(11,13)(12,15)(14,18)(16,17);(4,6)(7,8)(9,10)(19,20,22,24)(21,23,25,26);(1,2);(3,4)(5,6)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,25)(24,26);(11,12)(13,15)(14,17)(16,18)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72327586, C₂×#72327582, C₂×#72327608
maximal normal subgroup of	C₂×#72327586, C₂×#72327582, C₂×#72327608

GAP

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

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

C2×#5955311 #72339397