#7662076

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

6797 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)(5,6,10,11)(7,15,16,12)(8,13,18,17)(9,14,19,20)(21,22,26,24)(23,27,29,30)(25,28,33,32)(31,34,35,36);(1,2)(3,4)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(6,11)(7,16)(13,17)(14,20)(23,29)(27,30)(31,35)(34,36);(5,7,10,16)(6,12,11,15)(8,14,18,20)(9,17,19,13)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,8)(6,13)(7,14)(9,12)(10,18)(11,17)(15,19)(16,20)(21,24,26,22)(23,30,29,27)(25,28,33,32)(31,34,35,36);(5,9)(6,14)(7,17)(8,15)(10,19)(11,20)(12,18)(13,16)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=ghi,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7569272);