#478475

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

797 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)(7,17)(8,14,24,31)(9,15,10,13)(12,18)(19,34)(20,27)(21,26)(22,30,23,28)(25,35)(29,33)(32,36);(5,7)(6,12,16,27)(8,19)(9,20)(10,18)(11,21)(13,26,15,17)(14,29,31,25)(22,35)(23,33)(24,36)(28,32,30,34);(6,13)(7,18)(8,22)(9,10)(15,16)(17,26)(20,21)(23,24)(25,32)(28,30)(29,34)(33,35);(2,4)(5,8,9,22,11,24,10,23)(6,14,15,30,16,31,13,28)(7,19,20,35,21,36,18,33)(12,25,17,32,27,29,26,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385671);