#205567

order	512
nilpotency class	3
isoclinism	isoclinic class of #205565
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60639	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112763);