#8057873

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=gi,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, 7965069);