#476034

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,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,d^b=di,d^c=de,e^c=ei,f^c=fi,h^c=hi

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, 383230);