#8078322

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

4808 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)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29);(9,11,13,16)(10,14,12,15)(17,19,23,24)(18,21,20,26)(22,27,30,31)(25,29,28,32);(1,2,4,6)(3,5,7,8)(19,24)(21,26)(27,31)(29,32);(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,23,18)(19,21,24,26)(22,25,30,28)(27,32,31,29);(1,3,4,7)(2,5,6,8)(17,21)(18,24)(19,20)(22,29)(23,26)(25,31)(27,28)(30,32);(1,2)(3,5)(4,6)(7,8)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=h,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, 7985518);