#508549

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ei,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,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, 415745);