#7971683

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	#39372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4355 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)(9,10,13,14)(11,17,18,15)(12,16,21,20)(19,23,24,22)(26,31)(27,34)(33,37)(35,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,26,30,31)(27,32,34,28)(29,37,39,33)(35,36,40,38);(10,14)(11,18)(16,20)(19,24)(26,31)(28,32)(33,37)(36,38);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27)(26,32)(28,31)(29,35)(30,34)(33,38)(36,37)(39,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,40,37,35);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,39)(26,33,31,37)(27,35,34,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=gh,f^2=gh,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, 7878879);