#8075374

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

6494 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)(7,12,16,19)(8,9,13,18)(14,20,17,15)(21,22)(23,26)(24,28)(25,29)(27,31)(30,34)(32,33)(35,36);(1,2,3,4)(21,22)(23,27)(24,28)(25,29)(26,31)(30,35)(32,33)(34,36);(6,11)(9,18)(12,19)(15,20)(21,23,25,31)(22,26,29,27)(24,30,33,36)(28,34,32,35);(5,7)(6,12)(8,17)(9,15)(10,16)(11,19)(13,14)(18,20)(21,22)(23,26)(24,32)(25,29)(27,31)(28,33)(30,35)(34,36);(5,8,10,13)(6,9,11,18)(7,14,16,17)(12,20,19,15)(21,23)(22,27)(24,30)(25,31)(26,29)(28,35)(32,34)(33,36);(5,9)(6,13)(7,15)(8,11)(10,18)(12,14)(16,20)(17,19)(21,24)(22,28)(23,30)(25,33)(26,34)(27,35)(29,32)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982570);