#7743452

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

6648 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7650648);