#7812080

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	#45440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4435 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)(11,12)(13,18)(14,16)(15,17)(19,20)(21,23)(22,24)(25,26)(27,33)(28,32)(29,31)(30,35)(34,39)(36,38)(37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,16)(13,21)(14,17)(18,20)(19,22)(23,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(10,16)(11,17)(13,22)(20,23)(26,31)(30,33)(32,38)(37,39);(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,22)(23,24)(25,26)(27,30)(28,32)(29,31)(33,35)(34,37)(36,38)(39,40);(9,12)(10,17)(11,16)(13,20)(14,15)(18,21)(19,24)(22,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,18,16,24)(11,19,17,21)(12,20,15,23)(25,28,29,36)(26,32,31,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7719276);