#302989

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210185);