#7815865

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4570 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)(11,19,20,16)(12,18,21,24)(13,22,23,14);(1,2)(3,7)(4,8)(5,6)(9,11,15,20)(10,16,17,19)(12,13,21,23)(14,24,22,18)(25,26)(27,32)(28,33)(29,35)(30,31)(34,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(10,17)(11,20)(12,21)(14,22)(26,31)(29,39)(32,40)(33,36);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,13)(14,16)(15,21)(17,24)(19,22)(20,23)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(1,3,5,4)(2,7,6,8)(9,13)(10,14)(11,21)(12,20)(15,23)(16,24)(17,22)(18,19)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,4,5,3)(2,8,6,7)(9,14)(10,13)(11,18)(12,16)(15,22)(17,23)(19,21)(20,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7723061);