#8127016

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	C₂×#2404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5986 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=h,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^c=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, 8034212);