#418490

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325686);