#7714552

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	#51775	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 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)(13,21)(17,22)(18,24)(20,23)(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)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,31,29)(26,32,35,34)(28,36,39,38)(30,37,33,40);(10,15)(12,16)(17,22)(20,23)(25,28,31,39)(26,30,35,33)(27,36,29,38)(32,37,34,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,33)(27,36)(29,38)(30,35)(31,39)(32,40)(34,37);(1,3,5,4)(2,6,8,7)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,29,31,27)(26,34,35,32)(28,38,39,36)(30,40,33,37);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(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=gi,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, 7621748);