#348280

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

933 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255476);