#8075444

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4660 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(17,18,23,27)(19,28,29,24)(20,22,26,32)(21,30,31,25);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(17,19,23,29)(18,24,27,28)(20,21,26,31)(22,25,32,30);(1,2,5,8)(3,7,4,6)(9,12,13,10)(11,15,16,14)(19,29)(21,31)(24,28)(25,30);(1,3,5,4)(2,7,8,6)(17,20)(18,22)(19,21)(23,26)(24,25)(27,32)(28,30)(29,31);(1,4,5,3)(2,7,8,6)(17,21)(18,25)(19,26)(20,29)(22,28)(23,31)(24,32)(27,30);(1,2)(3,6)(4,7)(5,8)(17,22)(18,26)(19,25)(20,27)(21,28)(23,32)(24,31)(29,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=h,e^2=h,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, 7982640);