#110727

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

652 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efhi,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

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, 17923);