#405362

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1268 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)(5,6)(7,15)(8,12)(9,22)(10,14)(11,18)(13,25)(16,23)(17,33)(19,29)(20,30)(21,28)(24,27)(26,34)(31,35)(32,36);(1,2)(3,4)(5,6)(7,16,19,11)(8,12)(9,22)(10,14)(13,25)(15,23,29,18)(17,32,35,26)(20,30)(21,28)(24,27)(31,34,33,36);(1,3,4,2)(5,7)(6,11)(8,17)(9,23)(10,19)(12,26)(13,29)(14,16)(15,22)(18,25)(20,36)(21,35)(24,34)(27,31)(28,32)(30,33);(5,8,10,21)(6,12,14,28)(7,17,19,35)(9,24,25,20)(11,26,16,32)(13,30,22,27)(15,31,29,33)(18,36,23,34)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 312558);