#574144

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

2154 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)(7,11)(9,12)(13,14,17,20)(15,16,19,18)(21,22,26,27)(23,28,29,24)(25,30,33,34)(31,35,36,32);(1,2,4,3)(5,6,8,10)(7,9,11,12)(15,19)(16,18)(22,27)(23,29)(25,33)(32,35);(5,7)(6,9)(8,11)(10,12)(13,15,17,19)(14,18,20,16)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,34)(33,36);(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,16,17,18)(14,19,20,15)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(1,3,4,2)(5,7)(6,9)(8,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,30)(23,31)(24,32)(26,33)(27,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fh,d^2=i,e^2=hi,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, 481340);