#248420

order	512
nilpotency class	3
isoclinism	isoclinic class of #248359
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	#42480	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,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,d^c=di,f^c=fi,e^d=eh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155616);