#6444843

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#34889	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352039);