#7641996

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549192);