#9935987

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

5124 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

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, 9843183);