#8618171

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	#59029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4903 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8525367);