#248283

order	512
nilpotency class	3
isoclinism	isoclinic class of #248263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3109	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1073 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)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(18,23)(19,26)(20,21)(25,28)(27,29)(30,31);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(17,18)(19,24)(20,28)(21,23)(22,25)(26,30)(27,32)(29,31);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,18)(19,24)(20,23)(21,28)(22,25)(26,31)(27,32)(29,30);(17,19,22,27)(18,24,25,32)(20,29,21,26)(23,30,28,31)

Polycyclic presentation:

2^9|b^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fhi,f^c=fi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155479);