#395785

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1829 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)(7,16,19,13)(8,12)(9,14)(10,15)(11,18,26,23)(17,27,20,31)(21,28)(22,29)(24,30)(25,33,35,32)(34,36);(1,2)(3,4)(5,7)(6,11)(8,20)(9,23)(10,19)(12,25)(13,14)(15,26)(16,30)(17,22)(18,24)(21,32)(27,28)(29,35)(31,36)(33,34);(2,4)(5,8,9,21,10,22,24,34)(6,12,14,28,15,29,30,36)(7,17,18,32,19,20,23,33)(11,25,16,31,26,35,13,27);(5,7)(6,13)(8,17)(9,18)(10,19)(11,14)(12,27)(15,16)(20,22)(21,32)(23,24)(25,28)(26,30)(29,31)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=f,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,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, 302981);