#348014

order	512
nilpotency class	3
isoclinism	isoclinic class of #347998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29678	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

927 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:

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

Polycyclic presentation:

2^9|b^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ch,d^c=dh,f^c=fi,g^c=gi,f^d=fi

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, 255210);