#7651851

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	#42673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4752 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)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,18,22,21)(19,25,27,23)(20,24,30,29)(26,32,28,31);(1,3,5,8)(2,6,4,7)(10,14)(11,16);(9,11)(10,12)(13,16)(14,15)(18,21)(23,25)(24,29)(31,32);(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,28)(21,25)(22,27)(24,32)(26,30)(29,31);(17,20,22,30)(18,24,21,29)(19,26,27,28)(23,31,25,32);(1,4,5,2)(3,7,8,6)(17,21,22,18)(19,25,27,23)(20,29,30,24)(26,32,28,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7559047);