#325867

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233063);