#188890

order	512
nilpotency class	3
isoclinism	isoclinic class of #188824
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

991 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 991 conjugacy classes of subgroups — too many to draw. Only its 276 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(3,11)(7,15)(10,16)(12,13)(18,23)(20,27)(22,28)(26,31);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96086);