#394689

order	512
nilpotency class	3
isoclinism	isoclinic class of #394127
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	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301885);