C₂×#706362 #72325334

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

Aliases

direct product

C₂×#706362

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

40511 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)(7,8)(9,10)(13,17)(16,18)(19,20,23,26)(21,22,25,24);(3,4,5,6)(7,8)(9,10)(11,12,14,15)(13,18,17,16);(4,6)(12,15)(13,17)(19,21,23,25)(20,24,26,22);(4,6)(7,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,26)(23,25);(1,2);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,17)(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×#72232184, C₂×#72232135, C₂×#72232128
maximal normal subgroup of	C₂×#72232184, C₂×#72232135, C₂×#72232128

GAP

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

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

C2×#706362 #72325334