#576165

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2102 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=f,e^2=f,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, 483361);