#7812108

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

4390 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)(9,10,14,15)(11,16,17,23)(12,20,22,13)(18,19,21,24);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,17,23)(12,13,22,20)(18,24,21,19)(25,26)(27,32)(28,33)(29,35)(30,31)(34,39)(36,38)(37,40);(1,2,5,4)(3,6,8,7)(10,15)(11,17)(13,20)(18,21)(26,31)(29,39)(32,40)(33,36);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,21)(13,19)(14,17)(15,23)(18,22)(20,24)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(1,2,5,4)(3,6,8,7)(9,12)(10,13)(11,18)(14,22)(15,20)(16,24)(17,21)(19,23)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,4,5,2)(3,7,8,6)(9,13)(10,12)(11,19)(14,20)(15,22)(16,21)(17,24)(18,23)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,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, 7719304);