#7791025

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	#43010	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4580 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)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(1,3,5,4)(2,7,6,8)(10,17)(12,22)(16,19)(20,23)(26,30)(31,38)(32,35)(39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,28)(26,32)(27,33)(29,37)(30,35)(31,39)(34,36)(38,40);(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7698221);