#8001007

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

4689 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)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,30,31)(27,32,35,28)(29,37,38,33)(34,36,40,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,17)(18,19)(23,24)(26,31)(28,32)(33,37)(36,39);(11,20)(13,21)(15,19)(17,24)(25,27,30,35)(26,28,31,32)(29,34,38,40)(33,36,37,39);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,18,23)(11,21,20,13)(15,17,19,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,20)(21,24)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,22,20,12)(15,16,19,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7908203);