C₂×#4929208 #72325589

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

Aliases

direct product

C₂×#4929208

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

41737 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)(11,12)(13,17)(14,15)(16,18)(19,20,23,22)(21,25,26,24);(4,6)(8,9)(13,18)(16,17)(21,26)(24,25);(12,15)(13,18)(20,22)(21,26);(1,2);(7,9,10,8)(11,12)(13,17)(14,15)(16,18)(19,21)(20,24)(22,25)(23,26);(3,4)(5,6)(11,13,14,18)(12,16,15,17)(19,22,23,20)(21,24,26,25)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232147, C₂×#72232160, C₂×#72232170
maximal normal subgroup of	C₂×#72232147, C₂×#72232160, C₂×#72232170

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

C2×#4929208 #72325589