#8077413

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	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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)(11,15)(12,16)(13,23)(14,18)(17,21)(19,24)(20,22)(25,26,30,32)(27,28,34,36)(29,31,38,40)(33,35,37,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,21)(13,20)(14,19)(16,17)(18,24)(22,23)(25,27)(26,28)(29,37)(30,34)(31,39)(32,36)(33,38)(35,40);(1,3,5,4)(2,7,6,8)(11,19)(13,21)(15,24)(17,23)(25,28,30,36)(26,27,32,34)(29,35,38,39)(31,33,40,37);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,13)(14,20)(15,23)(17,24)(18,22)(19,21)(25,29)(26,31)(27,33)(28,35)(30,38)(32,40)(34,37)(36,39);(9,13)(10,17)(11,20)(12,19)(14,21)(15,16)(18,23)(22,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(9,10)(11,15)(12,22)(13,17)(14,18)(16,20)(19,24)(21,23)(25,27)(26,28)(29,37)(30,34)(31,39)(32,36)(33,38)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^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, 7984609);