C₂³×#3551 #72325005

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#3551

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

480781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#63041
maximal normal subgroup of	C₂³×#63041
maximal quotient of	C₂³×#47796, C₂³×#34415

GAP

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

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

C23×#3551 #72325005