#8091751

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	#46370	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4506 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)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18,23,27)(19,25,21,24)(20,22,26,32)(28,29,30,31);(9,10,12,13)(11,14,15,16)(17,19,23,21)(18,24,27,25)(20,30,26,28)(22,29,32,31);(2,5)(6,8)(19,21)(24,25)(28,30)(29,31);(3,7)(6,8)(17,20)(18,22)(19,28)(21,30)(23,26)(24,31)(25,29)(27,32);(1,3)(2,6)(4,7)(5,8)(17,21,23,19)(18,25,27,24)(20,28,26,30)(22,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,22)(18,26)(19,29)(20,27)(21,31)(23,32)(24,28)(25,30)

Polycyclic presentation:

2^9|a^2=i,b^2=g,e^2=i,f^2=gi,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, 7998947);