#466493

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 373689);