#403105

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1234 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)(5,6,9,25,11,13,10,15)(7,16,17,29,19,12,20,26)(8,21,24,28,22,35,23,14)(18,32,33,30,34,36,31,27);(5,7,11,19)(6,12)(8,18,22,34)(9,17,10,20)(13,16)(14,27)(15,29)(21,32)(23,31,24,33)(25,26)(28,30)(35,36);(6,13)(7,17)(9,10)(12,26)(14,28)(16,29)(18,31)(19,20)(23,24)(27,36)(30,32)(33,34);(2,4)(5,8)(6,14,13,28)(7,18)(9,23)(10,24)(11,22)(12,27,16,30)(15,21,25,35)(17,31)(19,34)(20,33)(26,36,29,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=efi,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=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, 310301);