#8080811

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

5093 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 7988007);