#8110544

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
prime radical	#36624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4880 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017740);