#8072285

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	#43096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 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)(10,14)(15,17)(16,22)(23,24)(25,26,31,30)(27,32,29,34)(28,33,37,36)(35,39,38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,27,31,29)(26,32,30,34)(28,35,37,38)(33,39,36,40);(10,14)(15,17)(16,22)(23,24)(26,30)(32,34)(33,36)(39,40);(9,11)(10,15)(12,20)(13,19)(14,17)(16,24)(18,21)(22,23)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,29,31,27)(26,34,30,32)(28,35,37,38)(33,39,36,40);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,30,31,26)(27,34,29,32)(28,36,37,33)(35,40,38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,f^2=gh,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, 7979481);