#480432

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1032 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,e^b=ei,f^b=fgi,d^c=di,f^e=fi

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, 387628);