#308924

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 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)(10,13)(14,16)(17,18)(19,30)(20,32)(21,25)(22,28)(23,27)(24,29)(26,31);(2,5)(6,7)(10,14)(12,15)(13,16)(17,19,23,20,24,31,22,21)(18,25,28,26,29,32,27,30);(9,10,11,13)(12,16,15,14)(17,20,22,19,24,21,23,31)(18,26,27,25,29,30,28,32);(1,3,4,8)(2,6,5,7)(17,21,22,31,24,20,23,19)(18,26,27,25,29,30,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 216120);