#7902538

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	#47838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4689 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)(11,18)(12,15)(19,23)(20,21)(25,26)(27,31)(28,39)(29,33)(30,34)(32,37)(35,38)(36,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,33)(34,35)(39,40);(9,11,14,18)(10,15,17,12)(13,19,22,23)(16,21,24,20)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,40)(37,39);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,28,30,37)(26,32,34,39)(27,36,38,29)(31,33,35,40);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,21)(14,15)(16,23)(17,18)(19,24)(20,22)(25,26)(27,31)(28,32)(29,40)(30,34)(33,36)(35,38)(37,39);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,24)(18,23)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,32,35,39)

Polycyclic presentation:

2^9|c^2=h,d^2=ghi,e^2=g,f^2=ghi,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, 7809734);