#320705

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227901);