#536371

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1914 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=gh,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, 443567);