#8073188

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

4844 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,e^2=g,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, 7980384);