#8075434

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	#59977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 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)(11,19)(12,17)(13,18)(14,16)(15,20)(21,22)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,23)(18,24);(10,16)(11,20)(13,17)(21,24)(25,27,30,28)(26,31,33,34)(29,35,37,36)(32,38,40,39);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,21)(13,16)(14,18)(15,23)(19,22)(20,24)(25,28,30,27)(26,31,33,34)(29,35,37,36)(32,39,40,38);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,23)(18,24)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39);(1,4,5,3)(2,8,6,7)(9,13,14,17)(10,18,16,12)(11,22,20,23)(15,24,19,21)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|c^2=hi,d^2=gh,e^2=gh,f^2=g,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, 7982630);