#7891203

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	#48563	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7798399);