#8091741

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#59412	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7998937);