#7712928

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	#47849	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) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7620124);