#420773

order	512
nilpotency class	3
isoclinism	isoclinic class of #417903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1876 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:

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

Polycyclic presentation:

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

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, 327969);