#8075526

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	#43662	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4573 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)(3,6,8,7)(10,14)(15,16)(17,18)(19,28)(20,22)(21,26)(23,25)(24,27)(29,30)(31,32);(9,10,13,14)(11,15,12,16)(17,19,24,23)(18,25,27,28)(20,31,21,29)(22,30,26,32);(2,4)(6,7)(17,20,24,21)(18,22,27,26)(19,29,23,31)(25,32,28,30);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,15,14,16)(17,21,24,20)(18,26,27,22)(19,29,23,31)(25,32,28,30);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,15,14,16)(17,22,24,26)(18,21,27,20)(19,30,23,32)(25,29,28,31);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)(17,23,24,19)(18,25,27,28)(20,31,21,29)(22,32,26,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7982722);