#7715536

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	#37559	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=h,f^2=i,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, 7622732);