#7720235

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

4394 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7627431);