#388006

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

761 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)(5,6,12,16,10,9,11,15)(7,13,20,28,19,18,26,31)(8,21,25,14,24,30,23,22)(17,32,33,29,34,36,35,27);(5,7,11,20,10,19,12,26)(6,13,15,28,9,18,16,31)(8,17,23,33,24,34,25,35)(14,29,21,36,22,27,30,32);(5,8,10,24)(6,14,9,22)(7,17)(11,25,12,23)(13,27)(15,30,16,21)(18,29)(19,34)(20,35)(26,33)(28,36)(31,32);(5,9)(6,10)(7,18,19,13)(8,22)(11,16)(12,15)(14,24)(17,29,34,27)(20,31,26,28)(21,25)(23,30)(32,33,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295202);