#7687335

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4554 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)(3,4)(5,8)(6,7)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30);(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30);(2,7)(3,8)(9,10,13,12)(11,14,16,15)(18,22)(23,29)(24,30)(31,32);(1,3,6,8)(2,4,7,5)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30);(1,4,6,5)(2,3,7,8)(9,12,13,10)(11,15,16,14)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31);(1,5,6,4)(2,8,7,3)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=hi,e^2=gh,f^2=h,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, 7594531);