#579695

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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

2430 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)(12,13)(14,16)(17,18)(19,21)(20,27)(22,23)(24,29)(25,30)(26,32)(28,31);(1,2,4,3)(6,8)(9,11,13,16)(10,14,12,15)(18,23)(19,25)(24,27)(26,31);(5,6)(7,8)(10,12)(14,15)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32);(1,3,4,2)(9,11)(10,14)(12,15)(13,16)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32);(9,12,13,10)(11,15,16,14)(17,21)(18,19)(20,28)(22,30)(23,25)(24,26)(27,31)(29,32)

Polycyclic presentation:

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

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, 486891);