#189307

order	512
nilpotency class	3
isoclinism	isoclinic class of #189298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

920 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^c=fhi,g^c=gh

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, 96503);