#418810

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60371	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1908 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:

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

Polycyclic presentation:

2^9|a^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,e^d=ei,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, 326006);