C₂³×#2749 #72296914

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

Aliases

direct product

C₂³×#2749

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

101837 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#3333823, C₂²×#1427668, C₂³×#57971, …
maximal normal subgroup of	C₂²×#3333823, C₂²×#1427668, C₂³×#57971, …

GAP

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

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

C23×#2749 #72296914