#7685478

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	#36182	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4807 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=gi,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, 7592674);