#7710923

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	#45482	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 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)(10,15)(13,17)(16,22)(19,23)(25,26)(27,31)(28,37)(29,38)(30,32)(33,36)(34,39)(35,40);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,16,20,22)(12,17,21,13)(18,23,24,19)(25,27,30,35)(26,31,32,40)(28,29,36,39)(33,34,37,38);(10,15)(12,21)(16,22)(18,24)(26,32)(27,35)(28,36)(34,38);(1,3,5,4)(2,7,6,8)(9,11,14,20)(10,16,15,22)(12,18,21,24)(13,19,17,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,37)(38,39);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,18)(13,15)(14,21)(16,23)(19,22)(20,24)(25,28,30,36)(26,33,32,37)(27,29,35,39)(31,34,40,38);(1,4,5,3)(2,8,6,7)(9,13,14,17)(10,12,15,21)(11,19,20,23)(16,18,22,24)(25,29)(26,34)(27,36)(28,35)(30,39)(31,37)(32,38)(33,40)

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618119);