#7654752

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

4980 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7561948);