#7966284

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	#59029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4861 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7873480);