#7699079

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	#52096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5156 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606275);