#428069

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×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1192 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)(7,16)(8,21)(9,11)(10,14)(12,19)(13,22)(15,26)(17,31)(18,20)(23,29)(24,30)(25,35)(27,33)(28,32)(34,36);(5,7,10,19)(6,12,14,16)(8,17)(9,25,15,18)(11,20,26,35)(13,28)(21,31)(22,32)(23,34)(24,33)(27,30)(29,36);(5,8)(6,13)(7,17)(9,21)(10,23)(11,24)(12,27)(14,30)(15,29)(16,28)(18,31)(19,34)(20,33)(22,26)(25,36)(32,35);(2,4)(5,9,26,14,10,15,11,6)(7,18,35,12,19,25,20,16)(8,21,22,30,23,29,24,13)(17,31,32,27,34,36,33,28)

Polycyclic presentation:

2^9|b^2=e,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, 335265);