#7664981

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	#37648	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5366 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572177);