#7644333

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

6472 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)(7,15)(8,10)(9,12)(11,14)(13,20)(16,17)(18,19)(21,22,26,25)(23,29,30,27)(24,32,33,28)(31,34,36,35);(1,2,3,4)(5,7,11,9)(6,12,14,15)(8,18,13,16)(10,17,20,19)(23,30)(27,29)(31,36)(34,35);(7,9)(12,15)(16,18)(17,19)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,8,11,13)(6,10,14,20)(7,16,9,18)(12,19,15,17)(21,24,26,33)(22,28,25,32)(23,31,30,36)(27,34,29,35);(5,9,11,7)(6,12,14,15)(8,18,13,16)(10,19,20,17)(21,22,26,25)(23,27,30,29)(24,28,33,32)(31,34,36,35);(5,10)(6,13)(7,17)(8,14)(9,19)(11,20)(12,16)(15,18)(21,25,26,22)(23,29,30,27)(24,32,33,28)(31,35,36,34)

Polycyclic presentation:

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

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, 7551529);