#473469

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

812 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380665);