#508217

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ei,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^c=de,f^c=fh,h^c=hi

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, 415413);