#8108348

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4822 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)(9,10,14,16)(11,15,13,12)(17,18,23,26)(19,21,29,24)(20,22,30,25)(27,32,31,28);(2,5)(6,8)(11,13)(12,15)(17,19,23,29)(18,24,26,21)(20,27,30,31)(22,28,25,32);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,12,16,15)(17,20,23,30)(18,25,26,22)(19,27,29,31)(21,28,24,32);(9,12,14,15)(10,11,16,13)(17,21,23,24)(18,19,26,29)(20,28,30,32)(22,31,25,27);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,15,16,12)(17,19)(18,24)(20,27)(21,26)(22,28)(23,29)(25,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,22)(18,20)(19,28)(21,31)(23,25)(24,27)(26,30)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=gh,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, 8015544);