#8088191

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

4732 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)(17,18,23,27)(19,24,29,31)(20,30,21,25)(22,26,28,32);(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,15,12,16)(17,19,23,29)(18,24,27,31)(20,22,21,28)(25,32,30,26);(2,3)(6,7)(17,20,23,21)(18,25,27,30)(19,28,29,22)(24,26,31,32);(2,3)(4,8)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,25,27,30)(19,22,29,28)(24,26,31,32);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,16,14,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31);(1,4)(2,6)(3,7)(5,8)(17,22)(18,26)(19,21)(20,29)(23,28)(24,25)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,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, 7995387);