#557320

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

1944 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)(6,8)(10,12)(14,18)(16,19)(21,22)(23,26)(24,29)(25,30)(27,32)(28,33)(31,34)(35,36);(7,11)(10,12)(14,18)(15,20)(22,23)(25,33)(27,29)(31,36);(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34);(5,7)(6,10)(8,12)(9,11)(13,15,17,20)(14,19,18,16)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36);(5,8,9,6)(7,12,11,10)(13,16,17,19)(14,20,18,15)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=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, 464516);