#7651216

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4658 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7558412);