#7671271

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

4776 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)(5,8)(6,7)(10,14)(15,16)(17,18)(19,27)(20,25)(21,30)(22,24)(23,28)(26,31)(29,32);(1,3,6,5)(2,4,7,8)(9,10,13,14)(11,15,12,16)(17,19,22,28)(18,23,24,27)(20,21,29,31)(25,26,32,30);(2,7)(4,8)(18,24)(19,28)(20,29)(26,30);(1,4,6,8)(2,3,7,5)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,32)(30,31);(1,3,6,5)(2,4,7,8)(17,20,22,29)(18,25,24,32)(19,21,28,31)(23,26,27,30);(1,5,6,3)(2,8,7,4)(9,12,13,11)(10,16,14,15)(17,21)(18,26)(19,29)(20,28)(22,31)(23,32)(24,30)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578467);