#7641898

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	#37554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4804 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)(10,13)(11,15)(17,18)(19,22)(20,21)(23,31)(24,26)(25,32)(27,28)(29,30);(1,2,5,6)(3,7,4,8)(10,13)(14,16)(17,19,24,21)(18,20,26,22)(23,27,32,29)(25,28,31,30);(9,10,12,13)(11,14,15,16)(17,20,24,22)(18,21,26,19)(23,28,32,30)(25,29,31,27);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21,24,19)(18,20,26,22)(23,29,32,27)(25,28,31,30);(1,4,5,3)(2,8,6,7)(17,22,24,20)(18,19,26,21)(23,30,32,28)(25,27,31,29);(17,23)(18,25)(19,27)(20,28)(21,29)(22,30)(24,32)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549094);