#456636

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1479 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363832);