#8107544

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

5007 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=hi,e^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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014740);