#419179

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 326375);