C₂³×#2855 #72339232

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂³×#2855

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

128853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂³×#39464, C₂³×#35077, C₂³×#36621, …
maximal normal subgroup of	C₂³×#39464, C₂³×#35077, C₂³×#36621, …

GAP

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

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

C23×#2855 #72339232