C₂³×#2644 #72301686

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂³×#2644

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

73493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 30 points:

(3,4);(5,6);(1,2)(3,4)(5,6);(7,8,10,11)(9,13,14,12)(15,16)(17,20)(18,19)(21,22)(25,29)(26,28);(8,11)(9,14)(16,18)(20,22)(23,24,27,30)(25,28,29,26);(7,8)(9,13)(10,11)(12,14)(15,17)(16,20)(18,22)(19,21)(23,25)(24,28)(26,30)(27,29);(7,9,10,14)(8,12,11,13)(15,18,19,16)(17,20,21,22)(23,26,27,28)(24,29,30,25)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#39148, C₂³×#42566, C₂³×#52988
maximal normal subgroup of	C₂³×#39148, C₂³×#42566, C₂³×#52988

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), (7,8,10,11)(9,13,14,12)(15,16)(17,20)(18,19)(21,22)(25,29)(26,28), (8,11)(9,14)(16,18)(20,22)(23,24,27,30)(25,28,29,26), (7,8)(9,13)(10,11)(12,14)(15,17)(16,20)(18,22)(19,21)(23,25)(24,28)(26,30)(27,29), (7,9,10,14)(8,12,11,13)(15,18,19,16)(17,20,21,22)(23,26,27,28)(24,29,30,25));

C23×#2644 #72301686