#302217

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

973 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=ehi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,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, 209413);