#7905448

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	#36581	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4424 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,e^2=h,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, 7812644);