#247809

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
prime radical	#60920	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|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,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, 155005);