#289661

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

711 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)(5,6,10,12,9,14,8,13)(7,11,15,20,16,18,17,19)(21,22,24,28,25,29,33,27)(23,26,32,34,30,36,31,35);(1,2,4,3)(5,7,8,17,9,16,10,15)(6,11,13,19,14,18,12,20)(21,23,24,32,25,30,33,31)(22,26,28,34,29,36,27,35);(6,12)(7,15)(8,10)(11,18)(13,14)(16,17)(22,27)(23,30)(24,33)(26,34)(28,29)(35,36);(1,3,4,2)(7,16)(11,18)(15,17)(19,20)(21,24,25,33)(22,28,29,27)(23,31,30,32)(26,35,36,34)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=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, 196857);