#489289

order	512
nilpotency class	4
isoclinism	isoclinic class of #488889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

722 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)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(18,23)(19,22)(20,27)(24,32)(25,31)(26,29)(28,30);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,18,21,25)(19,24,22,30)(20,28,29,32)(23,26,31,27);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,19,21,22)(18,24,25,30)(20,26,29,27)(23,28,31,32);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(18,25)(23,31)(24,30)(28,32)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287259, #72287350, #72287363, …
maximal normal subgroup of	#72287259, #72287350, #72287363, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396485);