#7771998

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	#49587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7679194);