#288020

order	512
nilpotency class	3
isoclinism	isoclinic class of #282171
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

701 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)(5,6,10,12,9,13,8,14)(7,11,15,20,17,18,16,19)(21,22)(23,29)(24,26)(25,28)(27,31)(30,35)(32,34)(33,36);(1,2,4,3)(5,7,8,16,9,17,10,15)(6,11,14,19,13,18,12,20)(22,26)(23,30)(25,32)(27,35)(29,36)(31,33);(6,12)(7,15)(8,10)(11,18)(13,14)(16,17)(21,23,24,31)(22,27,26,29)(25,33,32,30)(28,35,34,36);(1,3,4,2)(6,13)(11,18)(12,14)(19,20)(22,26)(23,31)(28,34)(30,33)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,c^2=i,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,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, 195216);