#7657879

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	#36430	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4780 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)(3,6)(4,8)(5,7)(10,13)(14,16)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,30)(31,32);(1,3,5,8)(2,6,7,4)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,27,28)(20,29,26,31)(22,30,25,32);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,19,23,21)(18,24,27,28)(20,29,26,31)(22,30,25,32);(1,4)(2,3)(5,6)(7,8)(17,20,23,26)(18,25,27,22)(19,29,21,31)(24,32,28,30);(17,21,23,19)(18,24,27,28)(20,31,26,29)(22,30,25,32);(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,31,28,29)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=hi,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, 7565075);