#7713888

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	#38950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4208 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)(12,20)(16,21)(18,24)(22,23)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(1,2,5,6)(3,7,4,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(10,14)(15,17)(16,21)(22,23)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621084);