#7975028

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	#40170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7882224);