#537090

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2532 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fgi,d^2=gh,e^2=f,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, 444286);