#8091755

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,e^2=i,f^2=gi,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, 7998951);