#7715754

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	#42999	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4541 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7622950);