#411713

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|d^2=h,e^2=i,g^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^b=ei,f^b=fi,g^b=gi,g^c=gi

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, 318909);