#7719038

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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)(11,19)(12,23)(13,18)(14,16)(15,22)(17,21)(20,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(10,16)(15,19)(17,23)(18,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(9,12,14,21)(10,17,16,23)(11,20,22,13)(15,24,19,18)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13,14,20)(10,18,16,24)(11,21,22,12)(15,23,19,17)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626234);