#9838200

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

5242 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9745396);