#416795

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 323991);