#10096978

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

4780 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,f^2=h,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, 10004174);