#8057912

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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)(9,10,16,20)(11,17,22,23)(12,13,14,18)(15,21,24,19)(25,26)(27,34)(28,36)(29,32)(30,33)(31,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,16,22)(10,17,20,23)(12,19,14,21)(13,15,18,24)(25,27)(26,29)(28,35)(30,32)(31,37)(33,34)(36,40)(38,39);(9,12,16,14)(10,18,20,13)(11,21,22,19)(15,23,24,17)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(9,13,16,18)(10,12,20,14)(11,15,22,24)(17,19,23,21)(25,28)(26,31)(27,35)(29,37)(30,38)(32,39)(33,36)(34,40);(9,14,16,12)(10,18,20,13)(11,19,22,21)(15,23,24,17)(25,29,30,34)(26,32,33,27)(28,37,38,40)(31,39,36,35);(1,3,4,7)(2,6,5,8)(9,15)(10,19)(11,18)(12,23)(13,22)(14,17)(16,24)(20,21)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965108);