#500734

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1059 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=ei,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,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, 407930);