#115001

order	512
nilpotency class	3
isoclinism	isoclinic class of #100227
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

867 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=di,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22197);