#10102574

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

4807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 10009770);