#7643457

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6,10,13)(7,11,15,8)(9,17,18,12)(14,16,20,19)(22,27)(23,25)(29,32)(31,33);(1,2,3,4)(5,7)(6,11)(8,13)(9,14)(10,15)(12,19)(16,17)(18,20);(5,8)(6,7)(9,16)(10,11)(12,14)(13,15)(17,20)(18,19)(21,22)(23,28)(24,32)(25,30)(26,27)(29,34)(31,36)(33,35);(21,23,26,25)(22,28,27,30)(24,31,34,33)(29,35,32,36);(5,7)(6,11)(8,13)(9,14)(10,15)(12,19)(16,17)(18,20)(21,24,26,34)(22,29,27,32)(23,31,25,33)(28,35,30,36);(5,9,10,18)(6,12,13,17)(7,14,15,20)(8,16,11,19)(21,25,26,23)(22,30,27,28)(24,33,34,31)(29,36,32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=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, 7550653);