#411292

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei

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, 318488);