#418155

order	512
nilpotency class	3
isoclinism	isoclinic class of #417487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1599 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325351);