#8105995

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

4896 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)(9,10,12,14)(11,15,16,13)(17,18,23,26)(19,24,20,21)(22,30,31,25)(27,29,28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,20)(18,21,26,24)(22,27,31,28)(25,29,30,32);(2,5)(6,8)(11,16)(13,15)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,32,30,29);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,21,23,24)(18,20,26,19)(22,29,31,32)(25,28,30,27);(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)

Polycyclic presentation:

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

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, 8013191);