#471155

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

776 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)(9,10,13,26,14,17,28,18)(11,15,29,31,20,32,21,37)(12,22,24,34,25,16,27,35)(19,30,38,33,36,40,23,39);(1,2,4,5)(3,6,7,8)(9,11,14,20)(10,15)(12,23)(13,21,28,29)(16,33,22,39)(17,32)(18,31)(19,27)(24,36)(25,38)(26,37)(30,34,40,35);(3,7)(6,8)(9,12)(10,16)(11,19)(13,27)(14,25)(15,30)(17,22)(18,35)(20,36)(21,38)(23,29)(24,28)(26,34)(31,39)(32,40)(33,37);(1,3,4,7)(2,6,5,8)(13,28)(16,22)(18,26)(19,36)(21,29)(24,27)(31,37)(33,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,d^2=ef,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,e^b=ei,f^b=fi,d^c=de,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378351);