#508347

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

611 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,h^c=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, 415543);