#9938133

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

4771 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gh,f^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, 9845329);