#434073

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1249 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)(5,6,10,15,11,14,9,25)(7,16,18,28,20,27,19,12)(8,21,22,13,24,36,23,30)(17,26,33,35,34,29,32,31);(5,7,9,19,11,20,10,18)(6,12,15,16,14,28,25,27)(8,17,22,33,24,34,23,32)(13,29,21,35,30,26,36,31);(5,8,11,24)(6,13)(7,17)(9,22,10,23)(12,26,28,29)(14,30)(15,21)(16,31,27,35)(18,32)(19,33)(20,34)(25,36);(2,4)(6,14)(7,18)(9,10)(12,16)(13,30)(17,32)(19,20)(22,23)(26,31)(27,28)(29,35)(33,34)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=efi,e^2=i,f^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=de,f^b=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341269);