#536615

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

2710 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^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, 443811);