#415456

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

778 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)(3,8,4,6)(9,10)(11,21)(12,17)(13,16)(14,30)(15,20)(18,24)(19,22)(23,38)(25,31)(26,37)(27,35)(28,36)(29,32)(33,39)(34,40);(1,3,5,4)(2,6,7,8)(9,11,22,29,15,24,14,13)(10,16,30,18,20,32,19,21)(12,23,26,39,28,40,27,25)(17,33,35,38,36,31,37,34);(1,4,5,3)(2,6,7,8)(9,12,22,26,15,28,14,27)(10,17,30,35,20,36,19,37)(11,23,29,39,24,40,13,25)(16,31,18,34,32,33,21,38);(9,13,14,24,15,29,22,11)(10,18,19,16,20,21,30,32)(12,25,27,40,28,39,26,23)(17,34,37,31,36,38,35,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322652);