#242627

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

770 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=di,d^c=dh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149823);