#559836

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333442, #72333467
maximal normal subgroup of	#72333442, #72333467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 467032);