#413470

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

986 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)(6,12)(7,16)(8,21)(9,10)(13,17)(14,29)(15,24)(18,32)(22,23)(25,28)(30,33)(31,35);(5,6,9,12,11,15,10,24)(7,17,20,13,16,25,19,28)(8,14,21,26,23,29,22,34)(18,31,33,27,30,35,32,36);(5,7,9,20,11,16,10,19)(6,13,12,25,15,28,24,17)(8,18,21,33,23,30,22,32)(14,27,26,35,29,36,34,31);(1,2,3,4)(5,8,9,21,11,23,10,22)(6,14,12,26,15,29,24,34)(7,18,20,33,16,30,19,32)(13,27,25,35,28,36,17,31)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,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,g^b=gi,e^c=ei,f^c=fi,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, 320666);