#7853381

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	#35364	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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)(11,12)(13,17)(14,18)(15,16)(19,20)(21,23)(22,24)(25,26,30,34)(27,33,38,31)(28,32,39,29)(35,40,37,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,15,18,12)(13,19,22,23)(17,21,24,20)(25,27)(26,31)(28,37)(29,36)(30,38)(32,40)(33,34)(35,39);(9,10,14,18)(11,12,16,15)(13,17,22,24)(19,20,23,21)(25,28)(26,32)(27,35)(29,34)(30,39)(31,36)(33,40)(37,38);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,23)(22,24)(25,29,30,32)(26,28,34,39)(27,36,38,40)(31,37,33,35);(9,12,14,15)(10,16,18,11)(13,21,22,20)(17,19,24,23)(25,28)(26,32)(27,37)(29,34)(30,39)(31,40)(33,36)(35,38);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,27)(26,33)(28,35)(29,36)(30,38)(31,34)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=h,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, 7760577);