#416438

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

920 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)(6,11)(7,15)(8,18)(9,20)(12,24)(13,28)(14,23)(16,30)(19,32)(21,27)(29,31)(33,35);(1,2,4,3)(5,6,9,11,10,14,20,23)(7,12,25,24,15,27,17,21)(8,13,18,22,19,28,32,34)(16,26,30,35,29,36,31,33);(1,3,4,2)(5,7,9,17,10,15,20,25)(6,12,11,21,14,27,23,24)(8,16,18,31,19,29,32,30)(13,26,22,33,28,36,34,35);(5,8,9,18,10,19,20,32)(6,13,11,22,14,28,23,34)(7,16,17,31,15,29,25,30)(12,26,21,33,27,36,24,35)

Polycyclic presentation:

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

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, 323634);