#3857044

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2167 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fh,c^2=fhi,d^2=hi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3764240);