#7656792

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	#42926	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4651 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=ghi,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, 7563988);