#8362084

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

4802 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8269280);