#191290

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#383
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 238 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)(3,11,14,15)(4,7)(5,9)(6,10)(8,13,16,12)(17,18,21,25,22,26,20,24)(19,23,29,32,27,30,28,31);(2,7,10,9)(8,15,16,11)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,32)(26,30);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(19,27)(23,30)(28,29)(31,32);(17,20,22,21)(18,24,26,25)(19,28,27,29)(23,31,30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,g^c=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98486);