#7714969

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

6732 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7622165);