#8908522

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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4338 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

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, 8815718);