#110795

order	512
nilpotency class	3
isoclinism	isoclinic class of #9519
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

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, 17991);