#7972299

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	C₂×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4606 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7879495);