#394596

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

2162 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi,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, 301792);