#366720

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2223 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6)(7,8)(9,10,14,20,16,21,15,19)(11,22,25,33,12,26,24,17)(13,28,30,36,31,35,29,18)(23,32,38,37,27,34,39,40);(1,2,3,4)(5,6)(7,8)(9,11,15,24,16,12,14,25)(10,17,19,26,21,33,20,22)(13,27,29,38,31,23,30,39)(18,34,35,37,36,32,28,40);(2,4)(6,8)(9,12)(10,17)(11,16)(13,27)(14,25)(15,24)(18,32)(19,22)(20,26)(21,33)(23,31)(28,40)(29,39)(30,38)(34,36)(35,37);(9,13,16,31)(10,18,21,36)(11,23,12,27)(14,30,15,29)(17,32,33,34)(19,35,20,28)(22,37,26,40)(24,39,25,38)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273916);