#417699

order	512
nilpotency class	3
isoclinism	isoclinic class of #417696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2324 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,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,f^c=fi,e^d=ei,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, 324895);