#290198

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

540 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)(9,10,13,11,14,16,12,15)(17,18,23,29,21,20,22,25)(19,26,30,31,27,24,28,32);(1,3,5,4)(2,6,8,7)(9,11,12,10,14,15,13,16)(17,19,23,30,21,27,22,28)(18,24,29,32,20,26,25,31);(9,10,14,16)(11,13,15,12)(17,20,21,18)(19,26,27,24)(22,25,23,29)(28,31,30,32);(1,4,5,3)(2,7,8,6)(18,20)(24,26)(25,29)(31,32)

Polycyclic presentation:

2^9|a^2=efhi,b^2=ef,c^2=h,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197394);