#7657903

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565099);