C₂³×#3539 #72324991

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

Aliases

direct product

C₂³×#3539

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:

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

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×#39844, C₂³×#36713, C₂³×#39862

GAP

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

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

C23×#3539 #72324991