#7731096

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	#38902	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4424 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)(27,34)(31,32)(35,40)(38,39);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26,30,28)(27,32,34,31)(29,36,37,33)(35,38,40,39);(1,3,5,4)(2,7,6,8)(10,14)(15,21)(16,19)(22,23)(25,27,30,34)(26,31,28,32)(29,35,37,40)(33,38,36,39);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,30,26)(27,31,34,32)(29,36,37,33)(35,38,40,39);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23)(25,29,30,37)(26,33,28,36)(27,35,34,40)(31,38,32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7638292);