#322500

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=egh,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,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, 229696);