#345113

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=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, 252309);