#7799523

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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)(9,10,15,14)(11,19,20,16)(12,17,22,23)(13,24,21,18);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,21)(13,22)(14,19)(15,20)(17,24)(18,23)(25,26)(27,32)(28,29)(30,31)(33,38)(34,40)(35,37)(36,39);(1,3,4,7)(2,6,5,8)(10,14)(11,20)(12,22)(18,24)(26,31)(29,33)(32,39)(35,40);(9,12,15,22)(10,17,14,23)(11,21,20,13)(16,24,19,18)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,40)(34,38);(9,13,15,21)(10,18,14,24)(11,12,20,22)(16,17,19,23)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(9,14,15,10)(11,19,20,16)(12,23,22,17)(13,24,21,18)(25,29,30,33)(26,28,31,38)(27,35,36,40)(32,37,39,34)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7706719);