#8421819

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	#37660	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^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, 8329015);