#7987491

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	#36122	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4736 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,12,15)(17,18)(19,25)(20,22)(21,31)(23,26)(24,27)(28,32)(29,30);(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19,24,21)(18,25,27,31)(20,30,23,28)(22,29,26,32);(2,3)(6,7)(17,20,24,23)(18,22,27,26)(19,28,21,30)(25,32,31,29);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,14,16,15)(17,21,24,19)(18,25,27,31)(20,30,23,28)(22,32,26,29);(1,3,5,2)(4,6,8,7)(9,12,13,11)(10,15,16,14)(17,22)(18,20)(19,29)(21,32)(23,27)(24,26)(25,30)(28,31);(1,4,5,8)(2,6,3,7)(17,23,24,20)(18,26,27,22)(19,30,21,28)(25,29,31,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=ghi,e^2=g,f^2=i,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, 7894687);