#6394042

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342836
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,d^2=i,e^2=fi,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301238);