C₂×#1082859 #72294550

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#1082859

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

37951 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)(11,12,14,16)(13,15,17,18)(19,20,22,25)(21,24,26,23);(4,6)(8,10)(11,13,14,17)(12,15,16,18)(19,20,22,25)(21,24,26,23);(7,8)(9,10)(19,21)(20,23)(22,26)(24,25);(11,13)(12,15)(14,17)(16,18)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,12)(13,15)(14,16)(17,18)(19,20)(21,23)(22,25)(24,26);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293937
maximal normal subgroup of	C₂×#72293937
maximal quotient of	C₂×#72293986

GAP

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

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

C2×#1082859 #72294550