#7710967

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	#2656	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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)(10,15)(13,21)(16,22)(19,24)(25,26)(27,32)(28,31)(29,37)(30,34)(33,38)(35,39)(36,40);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,16,20,22)(12,13,17,21)(18,19,23,24)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(10,15)(12,17)(16,22)(18,23)(25,26,30,34)(27,32,28,31)(29,33,38,37)(35,40,36,39);(1,3,4,7)(2,6,5,8)(9,11,14,20)(10,16,15,22)(12,18,17,23)(13,19,21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(9,12,14,17)(10,13,15,21)(11,18,20,23)(16,19,22,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(9,13)(10,17)(11,19)(12,15)(14,21)(16,23)(18,22)(20,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618163);