#394277

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2703 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)(5,6)(7,11)(8,18)(9,22)(10,14)(12,21)(13,24)(15,30)(16,32)(17,27)(19,35)(20,29)(23,28)(25,31)(26,34)(33,36);(1,2,3,4)(5,7,10,17)(6,11,14,27)(8,19)(9,16,24,34)(12,25)(13,26,22,32)(15,20)(18,30)(21,33)(23,31)(28,36)(29,35);(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,29)(14,28)(16,33)(17,19)(18,22)(21,24)(26,35)(27,36)(30,32)(31,34);(9,24)(13,22)(16,34)(18,29)(21,23)(26,32)(30,35)(31,33)

Polycyclic presentation:

2^9|b^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,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, 301473);