#556879

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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

1960 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)(7,10)(8,9)(11,12)(21,22,25,28)(23,26,29,32)(24,27,31,34)(30,33,35,36);(1,2,3,4)(6,9)(7,11)(14,17)(18,20)(23,29)(26,32)(30,35)(33,36);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,18,17,20)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,7,8,11)(6,10,9,12)(21,24,25,31)(22,27,28,34)(23,30,29,35)(26,33,32,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=h,e^2=g,b^a=bf,c^a=cg,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, 464075);