#418507

order	512
nilpotency class	3
isoclinism	isoclinic class of #418504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1717 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)(7,19,21,15)(8,22,26,31)(9,13,27,29)(10,17,11,16)(14,25,32,24)(20,28,35,36)(23,30,34,33);(1,2)(3,4)(5,7)(6,13)(8,20)(9,11)(10,27)(12,21)(14,28)(15,16)(17,19)(18,29)(22,30)(23,25)(24,34)(26,35)(31,33)(32,36);(2,4)(5,8)(6,14)(7,20)(9,23)(10,24)(11,25)(12,26)(13,28)(15,30)(16,22)(17,31)(18,32)(19,33)(21,35)(27,34)(29,36);(5,9,11,7,12,27,10,21)(6,15,17,29,18,19,16,13)(8,23,25,20,26,34,24,35)(14,30,31,36,32,33,22,28)

Polycyclic presentation:

2^9|a^2=i,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,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, 325703);