#8091747

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	#39444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4750 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^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, 7998943);