#328419

order	512
nilpotency class	3
isoclinism	isoclinic class of #326735
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×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235615);