#8088328

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#39458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,21)(22,24)(25,26)(27,32)(28,30)(29,36)(31,34)(33,39)(35,37)(38,40);(1,3,5,8)(2,6,4,7)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,24,22,23)(25,27)(26,32)(28,36)(29,30)(31,35)(33,38)(34,37)(39,40);(1,2,5,4)(3,6,8,7)(10,12)(13,21)(15,17)(18,22)(27,35)(29,39)(32,37)(33,36);(9,11)(10,15)(12,17)(13,22)(14,19)(16,24)(18,21)(20,23)(25,28,31,38)(26,30,34,40)(27,33,35,36)(29,32,39,37);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,24,22,23)(25,29)(26,33)(27,30)(28,37)(31,39)(32,38)(34,36)(35,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,23)(17,24)(19,22)(25,30)(26,28)(27,29)(31,40)(32,36)(33,37)(34,38)(35,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=hi,e^2=g,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, 7995524);