#8188098

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	#40493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8095294);