#7697329

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	C₂×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4661 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=gi,f^2=i,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, 7604525);