#247861

order	512
nilpotency class	3
isoclinism	isoclinic class of #247783
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1229 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)(7,13)(12,15)(14,16)(17,18)(19,24)(20,29)(21,26)(22,31)(23,27)(25,28)(30,32);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,11)(10,16)(17,19)(18,24)(20,22)(21,30)(23,28)(25,27)(26,32)(29,31);(2,8)(4,5)(7,11)(9,10)(13,14)(15,16)(17,20,23,30)(18,25,27,24)(19,21,28,22)(26,32,31,29);(2,9)(3,12)(4,5)(7,15)(8,10)(11,16)(17,21,23,22)(18,26,27,31)(19,20,28,30)(24,29,25,32)

Polycyclic presentation:

2^9|c^2=hi,d^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,d^b=di,e^b=eh,e^c=eh,f^c=fi,e^d=eh,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, 155057);