#7857889

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
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4680 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7765085);