#254065

order	512
nilpotency class	3
isoclinism	isoclinic class of #253987
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	#24180	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=ghi,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, 161261);