#7971822

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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)(11,17)(12,15)(13,18)(14,16)(19,24)(20,22)(21,23)(26,28)(27,34)(33,36)(35,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,26,30,28)(27,32,34,31)(29,36,37,33)(35,38,40,39);(10,16)(15,17)(18,23)(22,24)(25,27,30,34)(26,31,28,32)(29,35,37,40)(33,38,36,39);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(1,3,5,4)(2,6,8,7)(9,12,14,11)(10,17,16,15)(13,20,21,19)(18,24,23,22)(25,28,30,26)(27,31,34,32)(29,33,37,36)(35,39,40,38);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29,30,37)(26,33,28,36)(27,35,34,40)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,e^2=g,f^2=g,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, 7879018);