#535604

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 442800);