#8213864

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	#41983	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4674 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:

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

Polycyclic presentation:

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

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, 8121060);