#394719

order	512
nilpotency class	3
isoclinism	isoclinic class of #393975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1655 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 301915);