#7659944

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

4829 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)(19,25)(23,26)(27,31)(30,32);(1,2)(3,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32);(1,3)(2,5)(4,8)(6,7)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(1,2)(3,6)(4,7)(5,8)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,2)(3,5)(4,7)(6,8)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

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

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, 7567140);