#423269

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60644	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1586 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)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,25)(18,27)(19,28)(21,29)(23,30)(24,32)(26,31)(33,34)(35,36);(5,7,9,20,11,21,10,19)(6,12,15,17,16,28,14,29)(8,23,24,18,26,34,25,35)(13,30,31,36,32,33,22,27);(5,8)(6,13)(7,18,21,35)(9,25)(10,24)(11,26)(12,27,28,36)(14,31)(15,22)(16,32)(17,33,29,30)(19,34,20,23);(2,4)(6,14)(7,19)(9,10)(12,28)(13,31)(15,16)(18,34)(20,21)(22,32)(23,35)(24,25)(27,36)

Polycyclic presentation:

2^9|b^2=e,c^2=ef,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,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 330465);