#536482

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
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)(5,6,10,9)(7,12,11,8)(13,14,17,16)(15,18,19,20)(21,22,26,27)(23,28,30,25)(24,29,34,35)(31,36,32,33);(1,3,4,2)(15,19)(18,20)(22,27)(25,28)(29,35)(33,36);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,23,26,30)(22,28,27,25)(24,32,34,31)(29,33,35,36);(5,8,10,12)(6,11,9,7)(21,24,26,34)(22,29,27,35)(23,31,30,32)(25,33,28,36);(5,9,10,6)(7,12,11,8)(13,16,17,14)(15,20,19,18)(21,25,26,28)(22,30,27,23)(24,33,34,36)(29,32,35,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=f,d^2=fg,e^2=fgh,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, 443678);