#8088049

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	#59029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4560 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,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, 7995245);