C₂³×#15259 #72325752

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²×C₄

Aliases

direct product

C₂³×#15259

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3159928 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)(16,18)(21,25)(23,26);(11,12)(13,14)(15,16)(17,18);(3,4);(5,6);(7,8,9,10)(11,12,13,14)(15,16)(17,18)(19,20,22,24)(21,23,25,26);(1,2)(3,4)(5,6);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,23)(22,25)(24,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1418391, C₂³×#4312024
maximal normal subgroup of	C₂³×#1418391, C₂³×#4312024

GAP

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

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

C23×#15259 #72325752