#536748

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fhi,e^2=fg,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, 443944);