#7799563

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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)(25,26,30,33)(27,34,31,28)(29,32,37,39)(35,40,38,36);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,38)(39,40);(10,15)(13,17)(16,23)(19,24)(25,28,30,34)(26,31,33,27)(29,36,37,40)(32,38,39,35);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,28)(26,27)(29,36)(30,34)(31,33)(32,35)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,38)(39,40);(1,4,5,3)(2,8,6,7)(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(25,29,30,37)(26,32,33,39)(27,35,31,38)(28,36,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7706759);