#7852773

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4256 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)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(27,35)(29,37)(32,36)(33,38);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,26,31,28)(27,36,35,32)(29,33,37,38)(30,34,40,39);(1,3,5,4)(2,7,6,8)(10,12)(16,17)(18,20)(23,24)(25,27,31,35)(26,32,28,36)(29,30,37,40)(33,39,38,34);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,26)(27,32,35,36)(29,38,37,33)(30,39,40,34);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29)(26,33)(27,30)(28,38)(31,37)(32,39)(34,36)(35,40);(1,4,5,3)(2,8,6,7)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,33)(36,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7759969);