#395659

order	512
nilpotency class	3
isoclinism	isoclinic class of #394034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1846 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=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, 302855);