#556800

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4046 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)(11,14)(12,13)(15,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32);(2,4)(5,6,7,8)(10,13)(11,15)(19,25)(22,28)(26,31)(29,32);(1,2)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(9,10)(11,14)(12,13)(15,16)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32);(9,11,12,15)(10,14,13,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=fg,e^2=g,b^a=bf,c^a=cg,c^b=ch,d^b=dg,e^b=ef,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, 463996);