#588193

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2234 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)(6,10)(7,11)(13,14)(15,18)(16,17)(19,20)(21,22,26,29)(23,24,30,27)(25,33,34,28)(31,35,36,32);(1,3,4,2)(5,6)(7,8)(9,10)(11,12)(14,17)(18,20)(23,30)(24,27)(31,36)(32,35);(6,10)(7,11)(13,15,16,19)(14,18,17,20)(21,23)(22,27)(24,29)(25,31)(26,30)(28,35)(32,33)(34,36);(5,7,9,11)(6,8,10,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,27)(22,23,29,30)(25,32,34,35)(28,31,33,36);(5,8)(6,11)(7,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgh,d^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 495389);