#7647691

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	#37276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4642 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7554887);