#8087967

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4459 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 7995163);