#7815845

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

4494 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)(11,19)(12,13)(14,17)(15,18)(16,20)(21,24)(22,23)(25,26,31,37)(27,35,29,32)(28,34,39,33)(30,36,40,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,17)(13,21)(14,23)(15,20)(18,19)(22,24)(25,27,31,29)(26,32,37,35)(28,36,39,38)(30,33,40,34);(11,20)(14,21)(16,19)(17,24)(25,28,31,39)(26,33,37,34)(27,38,29,36)(30,35,40,32);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,13,18,23)(11,17,20,24)(14,16,21,19)(25,28)(26,34)(27,36)(29,38)(30,32)(31,39)(33,37)(35,40);(9,13)(10,12)(11,21)(14,20)(15,23)(16,17)(18,22)(19,24)(25,29,31,27)(26,35,37,32)(28,36,39,38)(30,33,40,34);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,17,18,24)(11,13,20,23)(12,16,22,19)(25,30)(26,36)(27,34)(28,32)(29,33)(31,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=h,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, 7723041);