#7799556

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

4496 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)(25,26)(27,34)(28,33)(29,31)(30,35)(32,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,27)(26,31)(28,37)(29,30)(32,40)(33,39)(34,35)(36,38);(10,15)(13,17)(16,23)(19,24)(25,28,30,38)(26,32,35,33)(27,36,29,37)(31,39,34,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,28)(26,33)(27,37)(29,36)(30,38)(31,39)(32,35)(34,40);(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,29,30,27)(26,34,35,31)(28,37,38,36)(32,40,33,39);(1,4,5,2)(3,7,8,6)(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(25,26)(27,31)(28,33)(29,34)(30,35)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=gi,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, 7706752);