#8088005

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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)(9,10,15,17)(11,19,12,16)(13,18,24,14)(20,21,22,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,17,19)(13,20,24,22)(14,21,18,23)(27,28)(32,34)(35,36)(39,40);(1,3,5,4)(2,7,6,8)(10,17)(14,18)(16,19)(21,23)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(9,12,15,11)(10,16,17,19)(13,20,24,22)(14,23,18,21)(25,28,31,27)(26,32,29,34)(30,35,38,36)(33,40,37,39);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,20)(12,22)(14,17)(15,24)(16,23)(19,21)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,14,15,18)(10,13,17,24)(11,21,12,23)(16,20,19,22)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=h,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, 7995201);