#8105955

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

4892 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)(17,18,23,26)(19,27,20,24)(21,32,25,22)(28,29,30,31);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,19,23,20)(18,24,26,27)(21,28,25,30)(22,29,32,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,24,26,27)(21,28,25,30)(22,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,25)(18,22,26,32)(19,28,20,30)(24,29,27,31);(9,10)(11,14)(12,13)(15,16);(1,3)(2,6)(4,7)(5,8)(17,22)(18,25)(19,29)(20,31)(21,26)(23,32)(24,30)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,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, 8013151);