C₂³×#40584 #72300223

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#40584

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2019080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7849701, C₂³×#7918389, C₂³×#7652795
maximal normal subgroup of	C₂³×#7849701, C₂³×#7918389, C₂³×#7652795
maximal quotient of	C₂³×#1259281

GAP

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

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

C23×#40584 #72300223