#394364

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
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	#60371	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2562 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)(7,14)(8,22)(9,24)(10,18)(11,17)(12,16)(13,20)(15,25)(19,33)(21,29)(23,28)(26,32)(27,31)(30,34)(35,36);(5,7,10,20,11,8,12,21)(6,13,16,14,17,29,18,22)(9,19,25,34,26,23,27,35)(15,28,24,30,31,33,32,36);(2,4)(5,8)(6,14)(7,11)(9,23)(10,21)(12,20)(13,18)(15,30)(16,29)(17,22)(19,26)(24,33)(25,35)(27,34)(28,32)(31,36);(5,9)(6,15)(7,19)(8,23)(10,25)(11,26)(12,27)(13,28)(14,30)(16,24)(17,31)(18,32)(20,34)(21,35)(22,36)(29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301560);