#500601

order	512
nilpotency class	4
isoclinism	isoclinic class of #500599
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

805 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407797);