#536038

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 443234);