C₂³×#2495 #72301689

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

Aliases

direct product

C₂³×#2495

Subgroups

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

72421 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);(7,8,11,9)(10,12,14,13)(15,16,19,18)(17,20,21,22)(23,24,28,27)(25,29,26,30);(10,14)(12,13)(16,18)(17,21)(23,25,28,26)(24,29,27,30);(7,9,11,8)(10,13,14,12)(15,17)(16,20)(18,22)(19,21)(23,26,28,25)(24,29,27,30);(7,10,11,14)(8,12,9,13)(15,18,19,16)(17,22,21,20)(23,27,28,24)(25,29,26,30)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#42566, C₂³×#59945, C₂³×#59220, …
maximal normal subgroup of	C₂³×#42566, C₂³×#59945, C₂³×#59220, …

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

C23×#2495 #72301689