#322569

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
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)(5,6,11,14,10,15,9,19)(7,16,17,13,8,12,18,20)(21,22,24,28,25,29,33,27)(23,26,32,35,31,34,30,36);(1,2,4,3)(5,7,10,8)(6,12,15,16)(9,18,11,17)(13,14,20,19)(21,23,24,32,25,31,33,30)(22,26,28,35,29,34,27,36);(5,8,10,7)(6,13,15,20)(9,18,11,17)(12,14,16,19)(22,27)(23,30)(24,33)(26,34)(28,29)(31,32);(1,3,4,2)(5,9,10,11)(6,14,15,19)(7,17,8,18)(12,13,16,20)(23,31)(26,34)(30,32)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

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, 229765);