#7708419

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

4957 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7615615);