#553243

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 460439);