#418436

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60423	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1700 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:

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

Polycyclic presentation:

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

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, 325632);