#8599653

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
prime radical	#35393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5372 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)(3,5)(4,6)(7,8)(17,18,23,27)(19,21,29,24)(20,30,28,25)(22,26,32,31);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(17,19,23,29)(18,24,27,21)(20,22,28,32)(25,26,30,31);(2,4)(3,7)(9,10,12,13)(11,14,15,16)(17,20,23,28)(18,25,27,30)(19,22,29,32)(21,31,24,26);(3,7)(5,8)(11,15)(14,16)(17,19)(18,24)(20,22)(21,27)(23,29)(25,26)(28,32)(30,31);(1,4,6,2)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,24)(18,19,27,29)(20,26,28,31)(22,25,32,30);(1,5)(2,3)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,22)(18,26)(19,28)(20,29)(21,25)(23,32)(24,30)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304020
maximal normal subgroup of	#72304020
maximal quotient of	#72266886, #72267028, #72267029, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506849);