#7846818

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	#42927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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)(9,10,15,14)(11,18,12,16)(13,21,22,17)(19,23,20,24)(25,26,31,29)(27,32,30,34)(28,33,37,38)(35,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,14,18)(13,19,22,20)(17,23,21,24)(25,27)(26,32)(28,35)(29,34)(30,31)(33,39)(36,37)(38,40);(10,14)(11,12)(17,21)(19,20)(27,30)(32,34)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,16,14,18)(13,19,22,20)(17,24,21,23)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(9,13,15,22)(10,17,14,21)(11,19,12,20)(16,23,18,24)(25,29,31,26)(27,34,30,32)(28,33,37,38)(35,39,36,40);(9,14,15,10)(11,16,12,18)(13,21,22,17)(19,23,20,24)(25,30,31,27)(26,34,29,32)(28,36,37,35)(33,40,38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7754014);