#8475673

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

5260 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)(3,8)(4,7)(5,6)(10,13)(14,16)(20,28)(24,29)(27,32)(30,31);(2,6)(3,7)(11,15)(14,16)(17,18,22,26)(19,21,25,23)(20,29,28,24)(27,31,32,30);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30);(2,6)(4,8)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31);(1,3,5,7)(2,4,6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29);(1,4)(2,7)(3,6)(5,8)(17,21)(18,25)(19,26)(20,30)(22,23)(24,32)(27,29)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382869);