#7920206

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4282 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)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,24,20,22)(25,26,30,34)(27,31,37,40)(28,32,36,39)(29,33,35,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,24,21,22)(25,27)(26,31)(28,35)(29,36)(30,37)(32,38)(33,39)(34,40);(10,16)(11,12)(18,21)(19,20)(25,28)(26,32)(27,29)(30,36)(31,33)(34,39)(35,37)(38,40);(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,22,21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,24,17,22)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,20,21,19)(25,29,30,35)(26,33,34,38)(27,36,37,28)(31,39,40,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=g,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, 7827402);