#479806

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387002);