#10411722

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

5085 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)(3,5)(4,6)(7,8)(9,10,14,12)(11,16,13,15)(17,18,23,27)(19,26,30,24)(20,22,21,25)(28,29,31,32);(2,3)(6,7)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,32)(26,27);(9,11,14,13)(10,15,12,16)(17,20,23,21)(18,25,27,22)(19,28,30,31)(24,29,26,32);(2,3)(4,8)(9,11,14,13)(10,16,12,15)(17,21,23,20)(18,22,27,25)(19,28,30,31)(24,29,26,32);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,16,13,15)(17,19)(18,26)(20,28)(21,31)(22,29)(23,30)(24,27)(25,32);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,16,12,15)(17,22)(18,20)(19,29)(21,27)(23,25)(24,31)(26,28)(30,32)

Polycyclic presentation:

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

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, 10318918);