#8458056

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

5079 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8365252);