#580924

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

2222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,4,3)(5,6,8,10)(7,9,11,12)(15,19)(16,18)(21,22,26,27)(23,25,29,34)(24,31,33,28)(30,36,35,32);(1,2,4,3)(7,11)(9,12)(13,14,17,20)(15,16,19,18)(22,27)(23,29)(28,31)(30,35);(5,6)(7,9)(8,10)(11,12)(13,15,17,19)(14,18,20,16)(21,23)(22,25)(24,30)(26,29)(27,34)(28,32)(31,36)(33,35);(1,3,4,2)(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,19)(17,20)(21,24,26,33)(22,28,27,31)(23,30,29,35)(25,32,34,36);(5,6)(7,9)(8,10)(11,12)(13,16,17,18)(14,19,20,15)(21,25)(22,29)(23,27)(24,32)(26,34)(28,35)(30,31)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fh,d^2=i,e^2=fh,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, 488120);