#7826305

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

4560 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)(11,12)(13,18)(14,16)(15,17)(19,20)(21,23)(22,24)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,16)(13,21)(14,17)(18,20)(19,22)(23,24)(25,27,30,34)(26,31,33,38)(28,35,32,39)(29,36,37,40);(1,3,5,4)(2,7,6,8)(10,16)(11,17)(13,22)(20,23)(27,34)(31,38)(35,39)(36,40);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,22)(23,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,16)(13,20)(14,15)(18,21)(19,24)(22,23)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,24)(17,21)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=gh,e^2=gh,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, 7733501);