#10360915

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

4934 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10268111);