#10529219

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

5417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10436415);