C₂×#2112564 #72294512

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

Aliases

direct product

C₂×#2112564

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

37699 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)(11,12,14,15)(13,16,18,17)(19,20)(21,24)(22,25)(23,26);(7,8,9,10)(11,12)(13,16)(14,15)(17,18)(21,26)(23,24);(4,6)(12,15)(16,17)(19,21,22,26)(20,23,25,24);(11,13,14,18)(12,16,15,17)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,12)(13,17)(14,15)(16,18)(19,20)(21,24)(22,25)(23,26);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293912
maximal normal subgroup of	C₂×#72293912
maximal quotient of	C₂×#72293936

GAP

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

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

C2×#2112564 #72294512