#8097753

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

4856 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)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(18,23)(24,30)(25,26)(31,32);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(18,23)(19,27)(25,26)(28,29);(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32);(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,31)(26,32);(1,3)(2,6)(4,8)(5,7)(17,20,22,21)(18,25,23,26)(19,28,27,29)(24,31,30,32);(1,2)(3,5)(4,7)(6,8)(17,21,22,20)(18,26,23,25)(19,29,27,28)(24,32,30,31)

Polycyclic presentation:

2^9|e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004949);