#498758

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₁₆
socle	C₂³	C₂³×C₈
prime radical	C₂×#127	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405954);