#322572

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

773 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)(5,6,8,12,9,13,17,11)(7,10,16,19,15,18,14,20)(21,22)(23,27)(24,28)(25,29)(26,31)(30,34)(32,36)(33,35);(1,2,4,3)(5,7,8,16,9,15,17,14)(6,10,12,19,13,18,11,20)(21,23,30,33,24,31,25,32)(22,26,34,36,28,27,29,35);(6,11)(7,14)(8,17)(10,18)(12,13)(15,16)(21,23,24,31)(22,27,28,26)(25,33,30,32)(29,35,34,36);(1,3,4,2)(7,15)(10,18)(14,16)(19,20)(22,28)(26,27)(29,34)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=hi,d^2=ehi,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, 229768);