#8517403

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

5999 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261985
maximal normal subgroup of	#72261985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424599);