#428070

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=ei,d^2=ef,f^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,g^b=gi,e^c=ei,f^c=fi,g^d=gi

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, 335266);