#422305

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

1211 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)(5,6,11,16,12,7,10,15)(8,19,22,17,24,20,23,13)(9,18,27,30,28,14,26,25)(21,33,34,29,36,32,35,31);(5,7,12,6)(8,20)(9,25,28,30)(10,15,11,16)(13,23)(14,26,18,27)(17,22)(19,24)(21,29)(31,36)(32,35)(33,34);(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,24)(14,29)(15,19)(16,20)(18,31)(25,33)(26,34)(27,35)(28,36)(30,32);(2,4)(5,9,10,26,12,28,11,27)(6,14,15,30,7,18,16,25)(8,21,22,34,24,36,23,35)(13,29,19,32,17,31,20,33)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=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, 329501);