#8069743

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

4873 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976939);