#324838

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

577 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232034);