#7901163

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	#47950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4608 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7808359);