#8063106

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

4661 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)(3,4)(5,6)(7,8)(10,14)(15,16)(17,18)(19,23)(20,25)(21,29)(22,26)(24,31)(27,30)(28,32);(1,3)(2,4)(5,7)(6,8)(9,10,13,14)(11,15,12,16)(17,19,22,29)(18,23,26,21)(20,28,31,27)(24,30,25,32);(3,7)(4,8)(9,11,13,12)(10,15,14,16)(17,20,22,31)(18,24,26,25)(19,27,29,28)(21,30,23,32);(2,6)(4,8)(9,11,13,12)(10,15,14,16)(17,21)(18,19)(20,30)(22,23)(24,27)(25,28)(26,29)(31,32);(1,4)(2,3)(5,8)(6,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,28)(21,30)(22,31)(23,32)(24,26)(27,29);(1,2)(3,4)(5,6)(7,8)(9,12,13,11)(10,16,14,15)(17,18)(19,23)(20,25)(21,29)(22,26)(24,31)(27,30)(28,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 7970302);