#572860

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

2592 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)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28);(1,2,4,3)(6,8)(9,11,13,16)(10,12,14,15)(19,24)(20,23)(27,32)(28,31);(5,6)(7,8)(10,14)(11,16)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28);(1,2,4,3)(9,11)(10,12)(13,16)(14,15)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27);(1,3,4,2)(9,12)(10,11)(13,15)(14,16)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fg,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=df,e^a=eg,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, 480056);