#7646564

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

6678 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:

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

Polycyclic presentation:

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

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, 7553760);