#260492

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 167688);