#7871681

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	#37827	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=hi,e^2=gh,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, 7778877);