#500847

order	512
nilpotency class	4
isoclinism	isoclinic class of #500843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24281	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1003 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:

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

Polycyclic presentation:

2^9|b^2=ehi,d^2=ehi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239838
maximal normal subgroup of	#72239838

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408043);