#7658642

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

6462 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565838);