#110670

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17866);