#7684296

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	#35927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5058 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591492);