#285601

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18585	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192797);