#7658923

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

6852 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)(7,13,17,12)(8,9,14,19)(15,18,20,16)(22,28)(24,34)(29,31)(32,35);(1,2)(3,4)(6,11)(8,14)(12,13)(15,20)(21,22)(23,31)(24,26)(25,29)(27,28)(30,34)(32,36)(33,35);(5,7)(6,12)(8,15)(9,16)(10,17)(11,13)(14,20)(18,19)(21,23,27,25)(22,29,28,31)(24,32,34,35)(26,33,30,36);(5,8,10,14)(6,9,11,19)(7,15,17,20)(12,16,13,18)(21,24)(22,30)(23,32)(25,35)(26,28)(27,34)(29,36)(31,33);(5,7)(6,13)(8,15)(9,18)(10,17)(11,12)(14,20)(16,19)(21,25,27,23)(22,29,28,31)(24,35,34,32)(26,33,30,36);(5,9)(6,14)(7,16)(8,11)(10,19)(12,20)(13,15)(17,18)(21,26,27,30)(22,24,28,34)(23,33,25,36)(29,32,31,35)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566119);