#7732812

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	#37446	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4662 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,e^2=hi,f^2=hi,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, 7640008);