#7736315

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×#3561	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6740 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)(21,22,26,28)(23,29,30,25)(24,32,35,27)(31,34,33,36);(1,2,3,4)(5,7,11,8)(6,12,15,16)(9,14,19,17)(10,18,20,13)(23,30)(25,29)(31,33)(34,36);(7,8)(9,19)(12,16)(13,18)(21,23)(22,25)(24,33)(26,30)(27,36)(28,29)(31,35)(32,34);(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)(21,24,26,35)(22,27,28,32)(23,31,30,33)(25,34,29,36);(5,10)(6,14)(7,13)(8,18)(9,12)(11,20)(15,17)(16,19)(21,25)(22,23)(24,34)(26,29)(27,31)(28,30)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 7643511);