#7989816

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

4451 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)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,31,33)(27,34,28,32)(29,35,36,30)(37,38,39,40);(1,2)(3,7)(4,8)(5,6)(12,19)(16,22)(18,23)(20,24)(25,27)(26,32)(28,31)(29,39)(30,40)(33,34)(35,38)(36,37);(10,15)(11,17)(16,22)(18,23)(26,33)(27,28)(29,36)(38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,31,27)(26,34,33,32)(29,39,36,37)(30,40,35,38);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,36)(26,35,33,30)(27,37,28,39)(32,40,34,38);(1,4,5,3)(2,8,6,7)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,30,31,35)(26,36,33,29)(27,38,28,40)(32,39,34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7897012);