#7709651

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	#38868	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4399 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26)(27,29)(28,36)(30,32)(31,34)(33,37)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(25,27,30,34)(26,31,32,29)(28,35,37,40)(33,39,36,38);(10,15)(11,16)(17,22)(18,23)(26,32)(27,34)(28,37)(38,39);(1,3,4,7)(2,6,5,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,37)(38,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29,30,31)(26,34,32,27)(28,38,37,39)(33,40,36,35)

Polycyclic presentation:

2^9|b^2=i,d^2=gh,f^2=h,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, 7616847);