#8424567

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4679 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8331763);