C₂×#534110 #72332529

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#534110

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

37725 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)(13,16)(14,15)(17,18);(7,8)(9,10)(12,15)(16,17)(22,26)(24,25);(12,15)(16,17)(19,20,23,21)(22,25,26,24);(1,2);(3,4,5,6)(11,12)(13,17)(14,15)(16,18)(19,21,23,20)(22,24,26,25);(11,13)(12,16)(14,18)(15,17)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323422, C₂×#72323412, C₂×#72323411, …
maximal normal subgroup of	C₂×#72323422, C₂×#72323412, C₂×#72323411, …

GAP

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

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

C2×#534110 #72332529