#7694893

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	#45269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4631 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,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, 7602089);