C₂×#2110383 #72325404

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

Aliases

direct product

C₂×#2110383

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

43017 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)(13,17)(14,18)(15,16)(19,20,23,21)(22,24,26,25);(4,6)(8,10)(12,16)(14,17)(19,20,23,21)(22,24,26,25);(8,10)(20,21)(24,25);(7,8,9,10)(11,13)(12,17)(14,16)(15,18)(19,21,23,20)(22,24,26,25);(1,2);(3,4,5,6)(11,14,15,17)(12,18,16,13)(19,22,23,26)(20,24,21,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232102, C₂×#72232143, C₂×#72232128
maximal normal subgroup of	C₂×#72232102, C₂×#72232143, C₂×#72232128

GAP

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

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

C2×#2110383 #72325404