#7736370

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
prime radical	C₂×#3558	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6992 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)(5,6,11,15)(7,16,8,12)(9,13,19,18)(10,14,20,17)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(1,2,3,4)(5,7,11,8)(6,12,15,16)(9,14,19,17)(10,18,20,13)(22,26)(24,32)(27,29)(30,36);(7,8)(9,19)(12,16)(13,18)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8,11,7)(6,12,15,16)(9,17,19,14)(10,18,20,13);(5,9)(6,13)(7,14)(8,17)(10,12)(11,19)(15,18)(16,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,10)(6,14)(7,13)(8,18)(9,12)(11,20)(15,17)(16,19)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643566);