#9935800

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

4808 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)(3,4)(5,8)(6,7)(10,14)(12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32);(1,3,7,8)(2,4,6,5)(9,10,13,14)(11,12,16,15)(18,23)(19,26)(20,28)(29,30);(1,2,7,6)(3,5,8,4)(9,10,13,14)(11,12,16,15);(1,4,7,5)(2,8,6,3)(9,11)(10,15)(12,14)(13,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,29,31,30);(1,5,7,4)(2,8,6,3)(9,11)(10,12)(13,16)(14,15)(17,20,22,28)(18,25,23,31)(19,27,26,32)(21,30,24,29);(1,6,7,2)(3,5,8,4)(9,12)(10,16)(11,14)(13,15)(17,21,22,24)(18,26,23,19)(20,29,28,30)(25,27,31,32)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 9842996);