C₂×#2077588 #72296400

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

Aliases

direct product

C₂×#2077588

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48517 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231820
maximal normal subgroup of	C₂×#72231820
maximal quotient of	C₂×#72278157

GAP

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

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

C2×#2077588 #72296400