C₂³×#2760 #72296897

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂³×#2760

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

112485 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:

(8,11)(9,13)(16,19)(17,21)(23,24)(25,28)(26,29)(27,30);(8,11)(12,14)(15,16,18,19)(17,20,21,22)(23,25)(24,27)(26,30)(28,29);(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,25)(24,28)(26,30)(27,29);(7,9)(8,12)(10,13)(11,14)(15,17,18,21)(16,20,19,22)(23,24)(25,27)(26,29)(28,30);(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×#4030235, C₂³×#57971
maximal normal subgroup of	C₂²×#4030235, C₂³×#57971

GAP

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

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

C23×#2760 #72296897