#8301428

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

5031 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)(3,4)(5,7)(6,8)(17,18,23,28)(19,24,29,31)(20,26,21,25)(22,27,32,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,28,31)(20,30,21,27)(22,25,32,26);(2,7)(4,6)(10,13)(14,16)(17,20,23,21)(18,25,28,26)(19,27,29,30)(22,31,32,24);(2,7)(3,8)(11,15)(14,16)(17,19)(18,24)(20,27)(21,30)(22,25)(23,29)(26,32)(28,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,26,28,25)(19,27,29,30)(22,31,32,24);(1,4,5,6)(2,3,7,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,27)(19,25)(20,24)(21,31)(23,32)(26,29)(28,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8208624);