#418536

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1673 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,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, 325732);