#7917230

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	#35682	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 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)(9,10,14,15)(11,13,18,16)(12,20,22,17)(19,21,24,23)(25,26,31,29)(27,32,38,35)(28,39,37,33)(30,34,36,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,22)(23,24)(25,27)(26,32)(28,36)(29,35)(30,37)(31,38)(33,34)(39,40);(1,3,5,4)(2,7,6,8)(10,15)(11,18)(17,20)(19,24)(25,26,31,29)(27,35,38,32)(28,33,37,39)(30,34,36,40);(1,4,5,3)(2,8,6,7)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,21,20,23)(25,28,31,37)(26,33,29,39)(27,36,38,30)(32,34,35,40);(9,12)(10,17)(11,19)(13,23)(14,22)(15,20)(16,21)(18,24)(25,29,31,26)(27,35,38,32)(28,39,37,33)(30,34,36,40);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,19,20,24)(25,30,31,36)(26,34,29,40)(27,37,38,28)(32,33,35,39)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=ghi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7824426);