#415685

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

814 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=gh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^c=gi,h^c=hi,f^d=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, 322881);