#322529

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, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eg,c^2=hi,d^2=ei,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, 229725);