C₂³×#16508 #72325809

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂³×#16508

Subgroups

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

1403944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 1024

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1259281, C₂³×#4312024
maximal normal subgroup of	C₂³×#1259281, C₂³×#4312024

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

C23×#16508 #72325809