#394634

order	512
nilpotency class	3
isoclinism	isoclinic class of #394034
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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2170 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:

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

Polycyclic presentation:

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

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, 301830);