#8088099

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

4933 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gi,e^2=g,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, 7995295);