#253358

order	512
nilpotency class	3
isoclinism	isoclinic class of #253339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18156	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,16)(14,15)(17,18,23,20)(19,26,27,32)(21,31,22,25)(24,29,30,28);(4,13)(6,15)(7,12)(11,14)(17,18,23,20)(19,26)(21,31,22,25)(24,28)(27,32)(29,30);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,19)(18,24)(20,30)(21,29)(22,28)(23,27)(25,32)(26,31);(3,10)(6,15)(9,16)(11,14)(17,20,23,18)(19,24,27,30)(21,25,22,31)(26,28,32,29)

Polycyclic presentation:

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

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, 160554);