C₂³×#3514 #72301662

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#3514

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

515957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#62983, C₂³×#63012, C₂³×#62912
maximal normal subgroup of	C₂³×#62983, C₂³×#63012, C₂³×#62912
maximal quotient of	C₂³×#39148, C₂³×#38910, C₂³×#59030, …

GAP

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

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

C23×#3514 #72301662