#8069686

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
prime radical	#59654	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^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, 7976882);