#348641

order	512
nilpotency class	3
isoclinism	isoclinic class of #348602
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=dh,f^c=fi,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255837);