C₂³×#3501 #72301663

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

Aliases

direct product

C₂³×#3501

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

503581 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);(7,8,11,13)(9,10,12,14)(15,16,19,21)(17,18,22,20);(7,8)(9,10)(11,13)(12,14)(15,17,19,22)(16,20,21,18);(7,9,11,12)(8,10,13,14)(15,18,19,20)(16,17,21,22);(7,10,11,14)(8,12,13,9);(7,8)(9,10)(11,13)(12,14)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#62983, C₂³×#62773
maximal normal subgroup of	C₂³×#62983, C₂³×#62773
maximal quotient of	C₂³×#59864, C₂³×#36154, C₂³×#36215

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), (7,8,11,13)(9,10,12,14)(15,16,19,21)(17,18,22,20), (7,8)(9,10)(11,13)(12,14)(15,17,19,22)(16,20,21,18), (7,9,11,12)(8,10,13,14)(15,18,19,20)(16,17,21,22), (7,10,11,14)(8,12,13,9), (7,8)(9,10)(11,13)(12,14));

C23×#3501 #72301663