#10320496

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

5141 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10227692);