#8125643

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5047 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gh,f^2=g,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, 8032839);