#7719417

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

4697 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)(4,7)(10,14)(15,16)(17,18,22,25)(19,21,27,23)(20,28,31,24)(26,32,29,30);(1,2,5,6)(3,4,8,7)(9,10,13,14)(11,15,12,16)(17,19)(18,21)(20,29)(22,27)(23,25)(24,32)(26,31)(28,30);(2,6)(3,8)(17,19,22,27)(18,23,25,21)(20,29,31,26)(24,30,28,32);(9,11,13,12)(10,15,14,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31);(1,3,5,8)(2,7,6,4)(17,20)(18,24)(19,26)(21,30)(22,31)(23,32)(25,28)(27,29);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,16,14,15)(17,21)(18,19)(20,30)(22,23)(24,26)(25,27)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626613);