#294382

order	512
nilpotency class	3
isoclinism	isoclinic class of #267755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1031 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,16,14,12)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(2,5)(6,7)(9,11,13,14)(10,12,15,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,25)(26,29)(31,32);(9,12,14,10,13,16,11,15)(17,20)(18,23)(19,26,27,28)(21,29)(22,30,25,32)(24,31);(1,3,4,8)(2,6,5,7)(10,15)(12,16)(18,24)(22,25)(23,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201578);