#7932356

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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)(9,10,15,14)(11,12,19,16)(13,21,22,17)(18,23,24,20)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,19)(10,16,14,12)(13,18,22,24)(17,23,21,20);(9,12,15,16)(10,11,14,19)(13,20,22,23)(17,18,21,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(1,3,5,4)(2,6,8,7)(9,12)(10,11)(13,20)(14,19)(15,16)(17,18)(21,24)(22,23)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,3,5,4)(2,6,8,7)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,4,5,3)(2,7,8,6)(9,14,15,10)(11,12,19,16)(13,17,22,21)(18,23,24,20)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7839552);