#7806828

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	#41404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4460 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7714024);