#8088318

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

7253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7995514);