#7658039

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

6610 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)(7,13)(11,14)(15,19)(18,20)(21,22,26,30)(23,31,34,27)(24,28,25,29)(32,36,33,35);(1,2)(3,4)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(21,22,26,30)(23,27,34,31)(24,28,25,29)(32,35,33,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,22)(23,27)(24,28)(25,29)(26,30)(31,34)(32,35)(33,36);(21,23,26,34)(22,27,30,31)(24,32,25,33)(28,35,29,36);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,35,31,36);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,25,26,24)(22,29,30,28)(23,33,34,32)(27,36,31,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=i,e^2=h,f^2=h,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, 7565235);