#8157142

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	#35005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4774 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)(3,5)(4,7)(6,8)(10,13)(11,15)(17,18,22,26)(19,27,29,23)(20,24,30,32)(21,31,28,25);(1,2,5,3)(4,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,22,29)(18,23,26,27)(20,28,30,21)(24,31,32,25);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32);(2,3)(6,7)(9,10)(11,14)(12,13)(15,16)(17,21)(18,25)(19,20)(22,28)(23,24)(26,31)(27,32)(29,30);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32);(1,4)(2,6)(3,7)(5,8)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064338);