#7656921

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

4823 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)(10,12)(11,15)(17,18)(19,28)(20,26)(21,25)(22,27)(23,30)(24,31)(29,32);(1,2)(3,7)(4,8)(5,6)(9,10,13,12)(11,16,15,14)(17,19,22,30)(18,23,27,28)(20,29,31,21)(24,32,26,25);(1,3,5,4)(2,7,6,8)(9,11,13,15)(10,14,12,16)(17,20)(18,24)(19,29)(21,30)(22,31)(23,32)(25,28)(26,27);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,16,15,14)(17,21,22,29)(18,25,27,32)(19,20,30,31)(23,24,28,26);(1,3,5,4)(2,7,6,8)(17,20)(18,26)(19,29)(21,30)(22,31)(23,25)(24,27)(28,32);(1,4,5,3)(2,8,6,7)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,30)(29,32)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=g,e^2=gh,f^2=gh,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, 7564117);