#7918526

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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)(10,14)(11,17)(16,22)(18,19)(25,26,32,31)(27,36,34,29)(28,33,40,39)(30,35,38,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,27)(26,29)(28,35)(30,33)(31,36)(32,34)(37,40)(38,39);(9,10,13,14)(11,15,17,21)(12,16,20,22)(18,23,19,24)(25,28)(26,33)(27,37)(29,38)(30,36)(31,39)(32,40)(34,35);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,29,32,36)(26,34,31,27)(28,38,40,30)(33,35,39,37);(9,11)(10,15)(12,19)(13,17)(14,21)(16,24)(18,20)(22,23)(25,30,32,38)(26,35,31,37)(27,33,34,39)(28,36,40,29);(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,31,32,26)(27,36,34,29)(28,39,40,33)(30,35,38,37)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825722);