#485107

order	512
nilpotency class	4
isoclinism	isoclinic class of #485065
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	#23988	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 910 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,26)(20,22)(21,25)(23,27)(24,29)(28,31)(30,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(18,22)(20,29)(23,30)(24,25)(26,31)(28,32);(19,27)(20,29)(23,26)(30,31);(2,7)(6,15)(8,12)(9,16)(17,19,21,27)(18,23,25,26)(20,28,29,32)(22,30,24,31)

Polycyclic presentation:

2^9|b^2=gi,d^2=i,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=di,e^c=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392303);