#10225571

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

5379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10132767);