#7981564

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

6777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7888760);