#238281

order	512
nilpotency class	3
isoclinism	isoclinic class of #238277
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,22,24,32)(13,17)(14,18)(16,25,31,26)(19,34,23,38)(20,29)(21,30)(27,33)(28,35,39,37)(36,40);(9,11)(10,15)(12,23,24,19)(13,21)(14,20)(16,28,31,39)(17,30)(18,29)(22,34,32,38)(25,37,26,35)(27,36)(33,40);(2,6)(3,4)(7,8)(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,32)(18,31)(20,23)(21,37)(22,33)(25,27)(29,39)(30,38)(34,40)(35,36);(1,3,5,4)(2,7,8,6)(11,20)(15,29)(19,23)(21,36)(28,39)(30,40)(34,38)(35,37)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=dh,g^b=gh,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319662
maximal normal subgroup of	#72319662

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145477);