#8075416

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

6636 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)(7,11)(8,10)(9,12)(13,15)(14,19)(16,17)(18,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,35)(32,34)(33,36);(1,2,3,4)(5,6,10,8)(7,11,15,13)(9,12,17,16)(14,19,18,20);(6,8)(9,17)(11,13)(14,18)(21,23,26,24)(22,27,29,30)(25,31,33,32)(28,34,36,35);(5,7,10,15)(6,11,8,13)(9,18,17,14)(12,20,16,19)(21,24,26,23)(22,27,29,30)(25,31,33,32)(28,35,36,34);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,20,18,19)(21,25,26,33)(22,28,29,36)(23,31,24,32)(27,34,30,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,19)(13,20)(15,18)(21,22)(23,30)(24,27)(25,28)(26,29)(31,35)(32,34)(33,36)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=h,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, 7982612);