#110804

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

770 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)(5,6,9,14,11,13,10,8)(7,15,18,20,17,12,19,16)(21,22,25,29,26,30,24,28)(23,31,34,35,32,27,33,36);(5,7,10,18,11,17,9,19)(6,12,8,16,13,15,14,20)(21,23)(22,27,30,31)(24,33)(25,34)(26,32)(28,35,29,36);(1,2,3,4)(5,8,10,13,11,14,9,6)(7,16,18,15,17,20,19,12)(22,28,30,29)(27,35,31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=i,f^2=hi,g^2=h,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, 18000);