#327777

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1140 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:

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

Polycyclic presentation:

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

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, 234973);