#418811

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:

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

Polycyclic presentation:

2^9|a^2=efi,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, 326007);