#488277

order	512
nilpotency class	4
isoclinism	isoclinic class of #487619
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2388	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

870 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395473);