#8075299

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

5002 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)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(17,18,24,26)(19,20,22,21)(23,25,31,32)(27,28,30,29);(9,11,13,16)(10,14,12,15)(17,19,24,22)(18,21,26,20)(23,27,31,30)(25,29,32,28);(2,5)(6,8)(17,20,24,21)(18,19,26,22)(23,29,31,28)(25,30,32,27);(1,3,4,7)(2,6,5,8)(17,21,24,20)(18,19,26,22)(23,28,31,29)(25,30,32,27);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,22,24,19)(18,20,26,21)(23,30,31,27)(25,28,32,29);(1,2)(3,6)(4,5)(7,8)(17,23)(18,25)(19,27)(20,28)(21,29)(22,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^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, 7982495);