#8657274

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

5744 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327643
maximal normal subgroup of	#72327643
maximal quotient of	#72327818, #72328541, #72329700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8564470);