#7910915

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	#42942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4515 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)(11,19)(12,21)(13,18)(14,16)(15,20)(17,22)(23,24)(25,26)(27,31)(28,36)(29,34)(30,32)(33,38)(35,40)(37,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,27,30,29)(26,31,32,34)(28,35,38,37)(33,39,36,40);(10,16)(11,20)(17,21)(18,24)(26,32)(31,34)(33,36)(39,40);(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40);(9,12,14,22)(10,17,16,21)(11,18,20,24)(13,19,23,15)(25,28,30,38)(26,33,32,36)(27,35,29,37)(31,39,34,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,17)(12,15)(14,23)(16,24)(19,22)(20,21)(25,29,30,27)(26,34,32,31)(28,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|b^2=gh,e^2=h,f^2=gh,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, 7818111);