#8057924

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,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, 7965120);