#8088057

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	#43109	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4585 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,f^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, 7995253);