#417730

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2411 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)(5,6,11,16)(7,17,21,34)(8,22,26,32)(9,15,10,14)(12,27,30,20)(13,24,33,25)(18,28,19,29)(23,35,36,31);(5,7)(6,12)(8,23)(9,27)(10,20)(11,21)(13,31)(14,34)(15,17)(16,30)(18,25)(19,24)(22,28)(26,36)(29,32)(33,35);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,28)(14,32)(15,22)(16,33)(17,35)(19,21)(20,36)(23,27)(29,30)(31,34);(5,8)(6,13)(7,19)(9,25)(10,24)(11,26)(12,29)(14,22)(15,32)(16,33)(17,35)(18,21)(20,36)(23,27)(28,30)(31,34)

Polycyclic presentation:

2^9|a^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 324926);