C₂×#3942867 #72325548

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#3942867

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

39889 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)(13,17)(14,18)(15,16)(21,25)(23,26);(11,13)(12,14)(15,18)(16,17)(19,20)(21,26)(22,24)(23,25);(7,8,9,10)(12,16)(13,18)(19,20,22,24)(21,23,25,26);(4,6)(7,8)(9,10)(11,13)(12,17)(14,16)(15,18)(19,20)(21,23)(22,24)(25,26);(1,2);(3,4)(5,6)(11,14,15,17)(12,13,16,18)(19,21,22,25)(20,23,24,26)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232181, C₂×#72232137, C₂×#72232165
maximal normal subgroup of	C₂×#72232181, C₂×#72232137, C₂×#72232165

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

C2×#3942867 #72325548