C₂³×#2778 #72296909

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂³×#2778

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)(11,14)(23,24)(25,29)(26,27)(28,30);(16,19)(17,21)(24,27)(25,30);(7,8,10,12)(9,11,13,14)(15,16,18,19)(17,22,21,20)(23,24)(25,28)(26,27)(29,30);(7,9)(8,11)(10,13)(12,14)(15,17,18,21)(16,20,19,22)(23,25,26,30)(24,28,27,29);(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×#2037594, C₂²×#1337713, C₂³×#57971
maximal normal subgroup of	C₂²×#2037594, C₂²×#1337713, C₂³×#57971

GAP

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

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

C23×#2778 #72296909