C₂³×#3521 #72324982

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

Aliases

direct product

C₂³×#3521

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

479709 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:

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

Conjugacy classes & character table

352 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×#43021, C₂³×#43530, C₂³×#59812

GAP

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

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

C23×#3521 #72324982