#7656421

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
prime radical	#36434	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7563617);