#576941

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2274 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=gh,e^2=f,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, 484137);