#8077888

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

5080 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)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,23,22)(19,26,21,24)(20,25,29,30)(27,32,28,31);(9,11,13,16)(10,14,12,15)(17,19,23,21)(18,24,22,26)(20,28,29,27)(25,32,30,31);(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,16,15)(18,22)(19,21)(25,30)(27,28);(1,3,4,7)(2,5,6,8)(9,10,13,12)(11,14,16,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32);(1,3)(2,5)(4,7)(6,8)(9,10,13,12)(11,14,16,15)(17,21,23,19)(18,26,22,24)(20,28,29,27)(25,32,30,31);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,15,16,14)(17,22,23,18)(19,26,21,24)(20,25,29,30)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7985084);