#322527

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)(21,22,25,28,26,29,24,30)(23,27,33,35,32,36,31,34);(1,2,4,3)(21,23,25,33,26,32,24,31)(22,27,28,35,29,36,30,34);(5,7,9,16)(6,11,12,19)(8,17,10,15)(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=fh,b^2=eg,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, 229723);