#247730

order	512
nilpotency class	3
isoclinism	isoclinic class of #247687
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	#60565	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1187 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(2,9)(4,5)(7,11)(8,10)(12,14)(15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154926);