#8057881

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

6203 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)(5,6,12,16)(7,13,18,19)(8,9,10,14)(11,17,20,15)(21,22)(23,29)(24,28)(25,27)(26,30)(31,35)(32,34)(33,36);(1,2)(3,4)(5,7,12,18)(6,13,16,19)(8,15,10,17)(9,11,14,20)(21,22,26,30)(23,27,25,29)(24,28,33,36)(31,34,32,35);(5,8,12,10)(6,14,16,9)(7,17,18,15)(11,19,20,13)(21,23,26,25)(22,27,30,29)(24,32,33,31)(28,35,36,34);(5,9,12,14)(6,8,16,10)(7,11,18,20)(13,15,19,17)(21,24,26,33)(22,28,30,36)(23,31,25,32)(27,34,29,35);(5,10,12,8)(6,14,16,9)(7,15,18,17)(11,19,20,13)(21,22)(23,27)(24,28)(25,29)(26,30)(31,34)(32,35)(33,36);(5,11)(6,15)(7,14)(8,19)(9,18)(10,13)(12,20)(16,17)(21,25,26,23)(22,29,30,27)(24,32,33,31)(28,35,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965077);