#8091753

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,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, 7998949);