#7902836

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

4444 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)(11,18)(12,15)(13,20)(14,17)(16,22)(19,23)(21,24)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,21)(22,24);(1,3,5,4)(2,7,6,8)(10,12)(11,17)(16,20)(19,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,21,24,23)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,3,5,4)(2,7,6,8)(9,13)(10,16)(11,19)(12,20)(14,23)(15,22)(17,24)(18,21)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,17,12,11)(13,21,22,23)(16,19,20,24)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7810032);