#8424580

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5054 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:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=gh,e^2=h,f^2=g,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, 8331776);