C₂³×#2712 #72300209

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄²

Aliases

direct product

C₂³×#2712

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

124197 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#51889, C₂³×#59803, C₂³×#43021, …
maximal normal subgroup of	C₂³×#51889, C₂³×#59803, C₂³×#43021, …
maximal quotient of	C₂³×#15314, C₂³×#15784

GAP

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

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

C23×#2712 #72300209